Ultralimits of pointed metric measure spaces
aa r X i v : . [ m a t h . M G ] F e b ULTRALIMITS OF POINTED METRIC MEASURE SPACES
ENRICO PASQUALETTO AND TIMO SCHULTZ
Abstract.
The aim of this paper is to study ultralimits of pointed metric measure spaces(possibly unbounded and having infinite mass). We prove that ultralimits exist under mildassumptions and are consistent with the pointed measured Gromov–Hausdorff convergence.We also introduce a weaker variant of pointed measured Gromov–Hausdorff convergence,for which we can prove a version of Gromov’s compactness theorem by using the ultralimitmachinery. This compactness result shows that, a posteriori, our newly introduced notionof convergence is equivalent to the pointed measured Gromov one. Another byproduct ofour ultralimit construction is the identification of direct and inverse limits in the category ofpointed metric measure spaces.
Contents
Chapter I. Introduction 31. General overview 32. Statement of results 6Chapter II. Preliminaries 103. Terminology about metric/measure spaces 104. Pointed measured Gromov–Hausdorff convergence 135. Ultrafilters and ultraproducts 176. Ultralimits of metric spaces 18Chapter III. Ultralimits of metric measure spaces 207. Construction of the ultralimit 208. On the support-concentration issue 289. Relation with the pmGH convergence 31Chapter IV. Weak pointed measured Gromov–Hausdorff convergence 3710. Definition of wpmGH convergence 3711. Relation between wpmGH convergence and ultralimits 3812. Equivalence between wpmGH convergence and pmG convergence 43Chapter V. Direct and inverse limits of pointed metric measure spaces 4613. The category of pointed metric measure spaces 46
Date : February 24, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Ultralimit, metric measure space, Gromov–Hausdorff convergence.
14. Direct limits of pointed metric measure spaces 4715. Inverse limits of pointed metric measure spaces 52Appendix A. Prokhorov theorem for ultralimits 56Appendix B. Tangents to pointwise doubling metric measure spaces 62HAPTER I
Introduction
It is well-known that in the uniformly doubling framework, the Gromov–Hausdorff limit ofa sequence of metric spaces is isometric to the ultralimit of that sequence. Such a relationbetween Gromov–Hausdorff convergence and ultralimit construction still exists in the settingof pointed metric spaces. Quite recently in [18], Elek generalised – using the so-called Loebmeasure construction – ultralimits to the context of bounded metric measure spaces with finitemeasure, and established an analogous connection between ultralimits and the (cid:3) -convergence`a la Gromov.In this paper we generalise the ultralimit construction to the setting where both the metricand the measure can be unbounded, and hence we consider pointed metric measure spaces.We also give a fairly complete picture of its connection to related notions of convergence. Moreprecisely, we prove that, under mild assumptions, pointed measured Gromov–Hausdorff limitsare isomorphic to the (support of) the ultralimit. Moreover, we prove that the isomorphismholds also for limits in the pointed measured Gromov topology introduced in [21]. In doingso, we study a weaker variant of pointed measured Gromov–Hausdorff convergence and provea version of Gromov’s compactness theorem in this setting. It turns out that in the end thisnotion of convergence is in fact equivalent to the pointed measured Gromov convergence.With the aid of ultralimits of metric measure spaces, we study direct and inverse limits ofsequences of pointed metric measure spaces, where the metric spaces under consideration arecomplete, separable, and equipped with boundedly finite Borel measures. More specifically,we will characterise when these limits exist in such category and explain how they look like.1.
General overview
The aim of this paper is to study ultralimits of metric measure spaces, as well as theirrelation with other important kinds of convergence, such as the measured Gromov–Hausdorffone and its variants. Before passing to the contents of the paper, we describe the motivationsbehind the different notions of convergence for spaces.1.1.
The role of metric measure spaces.
The notion of a ‘metric measure space’ certainlyplays a prominent role in many fields of mathematics, one of the reasons being that it is usedto provide natural generalisations of manifolds, where all types of singularities are allowed.At a purely metric level, important classes of spaces carrying a rich geometric structure arestudied. In this regard,
Alexandrov spaces constitute a significant example: introduced by A.
D. Alexandrov in [1], they are metric spaces where a synthetic lower bound on the sectionalcurvature is imposed via a geodesic triangles comparison. See [9] and the references therein.Nevertheless, other relevant concepts of curvature (or, rather, of curvature bounds) requirethe interplay between distance and ‘volume measure’ to be captured; in these situations, oneneeds to work in the category of metric measure spaces. This is the case, for instance, of lowerRicci curvature bounds: after the introduction of
Ricci limit spaces in the series of pioneeringpapers [11, 12, 13] by Cheeger–Colding in the nineties, the notion of a CD space (standing for C urvature- D imension condition) has been proposed by Lott–Villani [28] and Sturm [32, 33].The former are obtained by approximation from sequences of Riemannian manifolds satisfyinguniform lower Ricci bounds, while the latter is based on a purely intrinsic description of alower Ricci curvature bound. More precisely, the CD condition is expressed in terms of theconvexity properties of suitable entropy functionals in the Wasserstein space of probabilitymeasures having finite second-order moment; in particular, no smooth structure is involved inthe definition. It is then particularly evident that the nonsmooth theory of lower Ricci boundsmust be formulated in the setting of metric measure spaces, as the entropy functionals underconsideration depend both on the distance and the measure. Let us mention that the familyof CD spaces – differently from that of Ricci limits – contains also (non-Riemannian) Finslermanifolds. This observation led to the investigation of the so-called RCD condition , whichaims at selecting those ‘infinitesimally Hilbertian’ CD spaces that resemble a Riemannianmanifold. The class of RCD spaces has been widely studied as well, starting from [4, 3, 20].1.2.
Measured Gromov–Hausdorff convergence.
A crucial concept in metric geome-try is the
Gromov–Hausdorff distance , whose introduction dates back to [17, 22]. Roughlyspeaking, it is a distance on the ‘space of compact metric spaces’ that quantifies how far twometric spaces are from being isometric. When dealing with non-compact metric spaces, itis often useful to fix some reference points and to work with the pointed Gromov–Hausdorffconvergence .What often makes a notion of convergence worthwhile is its guarantee of stability and com-pactness for a suitable class of objects. In the framework of Gromov–Hausdorff convergence,the former refers to the fact that many classes of spaces – interesting from a geometric per-spective – are closed under Gromov–Hausdorff convergence; for instance, Alexandrov spaceshave this property, as observed by Grove–Petersen [24]. The latter means that every sequenceof spaces (where uniform bounds are imposed) subconverges to some limit space; in this direc-tion, the key result in the Alexandrov framework was achieved by Burago–Gromov–Perel’manin [10]. For a thorough account on Gromov–Hausdorff convergence, we refer to the monograph[9].When a measure enters into play – thus moving from metric spaces to metric measure spaces– one proper notion is that of (pointed) measured Gromov–Hausdorff convergence , introducedby Fukaya in [19] to study the behaviour of the eigenvalues of the Laplacian along a sequenceof Riemannian manifolds. The key idea behind the measured Gromov–Hausdorff convergenceis to couple the Gromov–Hausdorff distance with a suitable weak convergence of the involved
LTRALIMITS OF POINTED METRIC MEASURE SPACES 5 measures. On CD and RCD spaces, the measured Gromov–Hausdorff convergence turnedout to be extremely useful: indeed, both stability and compactness results have been provenin this setting. Amongst the other notions of convergence that are used within the lowerRicci bounds theory, we mention the D -convergence proposed by Sturm [32] and the pointedmeasured Gromov convergence by Gigli–Mondino–Savar´e [21]. Infinite-dimensional CD spacesneed not be locally compact and may be endowed with infinite ( σ -finite) reference measures,thus it would be quite unnatural to consider the pointed measured Gromov–Hausdorff distanceon them; the pointed Gromov convergence has been introduced exactly due to this reason.Notice that while the approach in the above mentioned notions differ from the approachin Gromov’s (cid:3) -convergence, there is still a natural connection between them (see e.g. [32, 27]and cf. [21, 18]).1.3. About ultralimits.
Another important approach to convergence in metric geometry isvia the ultralimit construction of a sequence of metric spaces, which aims at detecting the as-ymptotic behaviour of finite configurations of points in the approximating spaces. Ultralimitsgeneralise the concept of Gromov–Hausdorff convergence (as it was shown, for instance, in[26]) and rest on a given non-principal ultrafilter on the set of natural numbers. On the onehand, non-principal ultrafilters constitute a powerful technical tool, since they automaticallyprovide a simultaneous choice of sublimits. On the other hand, they cannot be explicitlydescribed, as their existence is equivalent to a weak form of the Axiom of Choice. We do notenter into the details of this discussion here, but we refer the interested reader to [23, 30, 7].We point out that also a ultralimit of measure spaces is available: it is the so-called
Loebmeasure space [16].Consequently, a natural question arises: is it possible to build ultralimits of metric measurespaces? An affirmative answer is given in the papers [15, 18] for finite measures, but it seemsthat in the existing literature the problem has not been explicitly investigated for (possiblyinfinite) σ -finite measures yet. Roughly, this is the plan we pursue in the present paper:1) We construct ultralimits of pointed metric measure spaces (under mild assumptions onthe measures under consideration), by suitably adapting the strategy used in [15, 18].2) We characterise when the ultralimit is ‘well-behaved’, in the sense that its measure isactually defined on the Borel σ -algebra; this point will become clearer in the sequel.3) We show consistency between ultralimits and measured Gromov–Hausdorff distance.4) We introduce the notion of weak pointed measured Gromov–Hausdorff convergence ,which is better suited for a comparison with ultralimits of metric measure spaces.5) We prove a compactness result for the weak pointed measured Gromov–Hausdorffconvergence by appealing to its connection with the ultralimits, in the spirit of theproof of the original Gromov’s compactness theorem. As a consequence, we obtain itsequivalence with the pointed measured Gromov convergence introduced in [21].6) We apply the ultralimit machinery to give new insights on direct and inverse limits in the category of pointed Polish metric measure spaces.In addition, we establish in the appendices the following: ENRICO PASQUALETTO AND TIMO SCHULTZ
7) We obtain a variant of Prokhorov theorem regarding ultralimits of measures.8) We point out the existence of weak pointed measured Gromov–Hausdorff tangents topointwise doubling spaces.Below we provide a more detailed account on the results that will be achieved in this paper.2.
Statement of results
Many of the definitions that we will adopt in this paper are not standard, thus let us beginby fixing some terminology.By a ball measure on a metric space (
X, d ) we mean a σ -finite measure m ≥ σ -algebra ( i.e. , the smallest σ -algebra containing all open balls). We will referto the triple ( X, d, m ) as a metric measure space . It is worth pointing out that if ( X, d )is separable, then ball measures and Borel measures on X coincide. The support of m isdefined as the set spt ( m ) of all points x ∈ X such that m (cid:0) B ( x, r ) (cid:1) > r >
0. Again,on separable spaces this notion coincides with the classical one. In general, the support isclosed and separable (see Lemma 3.2), but m is not necessarily concentrated on it. By ‘beingconcentrated’ we mean that m (cid:0) X \ spt ( m ) (cid:1) = 0. A subtle point here is that this identity iswell-posed because spt ( m ) belongs to the ball σ -algebra, as a consequence of Lemma 3.2.2.1. Ultralimits of metric measure spaces.
Fix a non-principal ultrafilter ω on N . Givena sequence (cid:0) ( X i , d i , p i ) (cid:1) of pointed metric spaces, we denote by ( X ω , d ω , p ω ) its ultralimit.See Sections 5 and 6 for a reminder of this terminology. In addition, suppose that each space( X i , d i ) is equipped with a ball measure m i . Then we prove – under a suitable volume growthcondition – that it is possible to construct the ultralimit of metric measure spaces ( X ω , d ω , m ω , p ω ) = lim i → ω ( X i , d i , m i , p i ) . The growth condition we consider is called ω -uniform bounded finiteness and states thatlim i → ω m i (cid:0) B ( p i , R ) (cid:1) < + ∞ , for every R > . See Definition 7.9. Under this assumption, we can build the sought ball measure m ω on X ω .Its construction – which follows along the lines of [18] and [15] – is technically involved, thuswe postpone its description directly to Section 7. We just mention that a key role in theconstruction is played by the class of internal measurable sets (see Definition 7.1), whichare defined as the ultraproducts of the measurable sets along the given sequence ( X i , d i , m i ).However, even if we start with a sequence of ‘very good’ spaces (say, compact metricspaces with uniformly bounded diameter and endowed with a Borel probability measure), theresulting ultralimit measure may still be extremely wild. The reason is that the ultralimit asa metric space can easily be non-separable, thus m ω might be a ‘truly’ ball ( i.e. , non-Borel)measure. A pathological instance of this phenomenon is described in Examples 7.16 and 8.4,where the ultralimit measure m ω is non-trivial but has empty support. In view of the aboveconsiderations, it is important to understand when the ultralimit measure is concentratedon its support; Section 8 will be devoted to this task. In these cases, m ω can be thought LTRALIMITS OF POINTED METRIC MEASURE SPACES 7 of as a Borel measure on a Polish space, since the support is always closed and separable.The philosophy is that, even when the ultralimit metric space is extremely big, the ultralimitmeasure may be capable of selecting a ‘more regular’ portion of the space. More specifically,in Theorem 8.3 we will provide necessary and sufficient conditions for the measure m ω to besupport-concentrated. The most relevant – which will appear again later in this introduction– is what we will call asymptotic bounded m ω -total boundedness in Definition 8.2: ∀ R, r, ε > ∃ M ∈ N , ( x in ) Mn =1 ⊂ X i : lim i → ω m i (cid:0) ¯ B ( p i , R ) \ S Mn =1 B ( x in , r ) (cid:1) ≤ ε. (2.1)Finally, in Section 9 we will investigate the relation between the pointed measured Gromov–Hausdorff convergence (recalled in Section 4) and the newly introduced notion of ultralimit.In this case, we consider pointed Polish metric measure spaces (meaning that the under-lying metric space is complete and separable), whose reference measure is boundedly-finite.Our main result here is Theorem 9.4, where we show that if a sequence (cid:0) ( X i , d i , m i , p i ) (cid:1) ofpointed Polish metric measure spaces converges to some ( X ∞ , d ∞ , m ∞ , p ∞ ) in the pointedmeasured Gromov–Hausdorff sense, then the ultralimit measure m ω is support-concentratedand the limit ( X ∞ , d ∞ , m ∞ , p ∞ ) can be canonically identified with (cid:0) spt ( m ω ) , d ω , m ω , p ω (cid:1) .2.2. Weak pointed measured Gromov–Hausdorff convergence.
By looking at theproof of the above-mentioned consistency result (namely, Theorem 9.4), one can realise thatit is possible to weaken the notion of pointed measured Gromov–Hausdorff convergence, butstill maintaining a connection with ultralimits. To be more precise, let us observe that insteadof working with (
R, ε ) -approximations (recalled in Definition 4.1), we can consider whatwe call weak ( R, ε ) -approximations (introduced in Definition 10.1). The difference is thatfor a weak ( R, ε )-approximation the ‘quasi-isometry’ assumption is required to hold up to asmall measure set . It seems that this variant better fits into the framework of metric measurespaces, because in its formulation distance and measure cannot be decoupled (differently fromwhat happens in the pointed measured Gromov–Hausdorff case, where to a ‘purely metric’concept of quasi-isometry, a control on the relevant measures is imposed only afterward).Once our notion of weak pointed measured Gromov–Hausdorff convergence is in-troduced in Definition 10.3, we can investigate its relation with ultralimits in Section 11. InTheorem 11.1 we prove that if a sequence of pointed Polish metric measure spaces is asymp-totically boundedly m ω -totally bounded (2.1), then it subconverges to (cid:0) spt ( m ω ) , d ω , m ω , p ω (cid:1) with respect to the weak pointed measured Gromov–Hausdorff convergence. This result maybe regarded as a reinforcement of Theorem 9.4, as it readily implies that each weak pointedmeasured Gromov–Hausdorff limit must coincide with the ultralimit (Corollary 11.3). Hav-ing these tools at our disposal, we can then easily obtain a Gromov’s compactness theorem(see Theorem 11.4) for weak pointed measured Gromov–Hausdorff convergence, stating thefollowing: a given sequence of spaces (cid:0) ( X i , d i , m i , p i ) (cid:1) is precompact with respect to the weak ENRICO PASQUALETTO AND TIMO SCHULTZ pointed measured Gromov–Hausdorff convergence if and only if it is boundedly measure-theoretically totally bounded , which means that ∀ R, r, ε > ∃ M ∈ N , ( x in ) Mn =1 ⊂ X i : sup i ∈ N m i (cid:0) ¯ B ( p i , R ) \ S Mn =1 B ( x in , r ) (cid:1) ≤ ε. A significant consequence of Theorem 11.4 is the equivalence between weak pointed measuredGromov–Hausdorff convergence and pointed measured Gromov convergence, that we willachieve in Theorem 12.2. In particular, by combining it with the results proven in [21], wecan conclude that for any K ∈ R the classes of CD ( K, ∞ ) and RCD ( K, ∞ ) spaces are closedunder weak pointed measured Gromov–Hausdorff convergence.2.3. Direct and inverse limits of pointed metric measure spaces.
In Chapter V weshow how ultralimits of pointed metric measure spaces can be helpful in constructing directand inverse limits in the category of pointed Polish metric measure spaces. We address theproblem in a quite restricted setting: first, we just consider inverse and direct limits, notother limits or colimits in the category; second, instead of an arbitrary directed index set( I, ≤ ), we just stick to the case of natural numbers N endowed with the natural order (inaccordance with the rest of the paper). A word on terminology: in Chapter V we considerpointed metric measure spaces ( X, d, m , p ) where p ∈ spt ( m ), in order to get a reasonablenotion of morphism (see Definition 13.1); this assumption is not in force in the rest of thepaper, as it would be quite unnatural on non-separable spaces, where non-trivial measuresmay have empty support.In Theorem 14.1 and Corollary 14.2 we characterise which are the direct systems of pointedPolish metric measure spaces admitting direct limit: they are exactly the uniformly boundedlyfinite ones. Moreover, in this case the sequence has a weak pointed measured Gromov–Hausdorff limit, which also coincides with the direct limit itself.The inverse limit case is more complex: given an inverse system of pointed Polish metricmeasure spaces (cid:0) ( X i , d i , m i , p i ) (cid:1) , one can select a subspace X of its ultralimit X ω having theproperty that the inverse limit exists if and only if p ω ∈ spt ( m ω x X ). Moreover, in this casethe inverse limit coincides with X equipped with the restricted distance and measure; seeTheorem 15.1. The space X – which is defined in (15.2) – a priori depends on the choice ofthe non-principal ultrafilter ω : its independence is granted by the uniqueness of the inverselimit. Actually, differently from what happens for direct limits, we are currently unaware ofa characterisation of the inverse limit that does not appeal to the theory of ultralimits.In the context of metric measure space geometry, inverse limits have been considered – forinstance – in the paper [14].2.4. Tangents to pointwise doubling metric measure spaces.
An essential tool inmetric measure geometry is given by the notion of a tangent cone . For instance, in thesetting of finite-dimensional
RCD spaces, where the study of fine structural properties of thespaces aroused a great deal of attention, tangent cones play a central role. Roughly speaking,given a pointed metric measure space (
X, d, m , p ), what we call a ‘tangent cone’ at p is anylimit of the rescaled spaces ( X, d/r i , m pr i , p ), where m pr i are suitable normalisations of m , and LTRALIMITS OF POINTED METRIC MEASURE SPACES 9 r i ց
0. In the case of finite-dimensional
RCD spaces, limits are taken with respect to thepointed measured Gromov–Hausdorff distance. We do not enter into details and we refer to[31, 8].In Appendix B we observe (see Theorem B.5) that on pointwise doubling metric measurespaces, at almost every point a weak pointed measured Gromov–Hausdorff tangent cone exists.By pointwise doubling (or asymptotically doubling, or infinitesimally doubling) we mean thatlim r ց m (cid:0) B ( x, r ) (cid:1) m (cid:0) B ( x, r ) (cid:1) < + ∞ , for m -a.e. x ∈ X. In particular, all doubling spaces have this property, but the class of pointwise doubling spacesis much larger and very well-studied in the framework of analysis on metric spaces.It is worth mentioning that Theorem B.5 applies also to some infinite-dimensional
RCD spaces, such as the Euclidean space endowed with a Gaussian measure. Anyway, we believethat this property is well-known to the experts (in its equivalent formulation using the pointedmeasured Gromov convergence) and thus we will not investigate further in that direction.
Acknowledgements.
The authors would like to thank the Department of Mathematics andStatistics of the University of Jyv¨askyl¨a, where much of this work was done. Both authorswere supported by the Academy of Finland, project number 314789. The first named authorwas also supported by the European Research Council (ERC Starting Grant 713998 GeoMeGGeometry of Metric Groups) and by the Balzan project led by Luigi Ambrosio. The secondnamed author was also supported by the Academy of Finland, project number 308659.HAPTER II
Preliminaries Terminology about metric/measure spaces
Let us begin by fixing some general terminology about metric and measure spaces, whichwill be used throughout the whole paper.3.1.
Metric spaces. An extended pseudodistance on a set X is a function d : X × X → [0 , + ∞ ] such that the following properties hold: d ( x, x ) = 0 , for every x ∈ X,d ( x, y ) = d ( y, x ) , for every x, y ∈ X,d ( x, y ) ≤ d ( x, z ) + d ( z, y ) , for every x, y, z ∈ X. The couple (
X, d ) is said to be an extended pseudometric space . An extended pseudodistancesatisfying d ( x, y ) < + ∞ for every x, y ∈ X is said to be a pseudodistance on X and the couple( X, d ) is a pseudometric space . A pseudodistance such that d ( x, y ) > x, y ∈ X are distinct is called a distance on X and the couple ( X, d ) is a metric space . Later on, weshall need the following two well-known results; we omit their standard proofs. • Let (
X, d ) be an extended pseudometric space. Let ¯ x ∈ X be given. Let us define O ¯ x ( X ) := (cid:8) x ∈ X (cid:12)(cid:12) d ( x, ¯ x ) < + ∞ (cid:9) . Then (cid:0) O ¯ x ( X ) , d (cid:1) is a pseudometric space. (For the sake of brevity, we wrote d insteadof d | O ¯ x ( X ) × O ¯ x ( X ) . This abuse of notation will sometimes appear later in this paper.) • Let (
X, d ) be a pseudometric space. Let ∼ be the following equivalence relation onthe set X : given any x, y ∈ X , we declare that x ∼ y provided d ( x, y ) = 0. Denoteby ˜ X the quotient set X/ ∼ . We define the function ˜ d : ˜ X × ˜ X → [0 , + ∞ ) as˜ d (˜ x, ˜ y ) := d ( x, y ) , for every ˜ x, ˜ y ∈ ˜ X, where x ∈ X (resp. y ∈ X ) is any representative of ˜ x (resp. ˜ y ). It is easy to checkthat the definition of ˜ d is well-posed, i.e. , it does not depend on the specific choice ofthe representatives x and y . Then it holds that ( ˜ X, ˜ d ) is a metric space.Given a pseudometric space ( X, d ), a point x ∈ X , and a real number r >
0, we denote B ( x, r ) := (cid:8) y ∈ X (cid:12)(cid:12) d ( x, y ) < r (cid:9) , ¯ B ( x, r ) := (cid:8) y ∈ X (cid:12)(cid:12) d ( x, y ) ≤ r (cid:9) . We call B ( x, r ) (resp. ¯ B ( x, r )) the open ball (resp. closed ball ) of center x and radius r . Moregenerally, given any A ⊂ X and a real number r >
0, we call A r the open r -neighbourhood ofthe set A , namely A r := (cid:8) x ∈ X : dist( x, A ) < r (cid:9) , where dist( x, A ) := inf y ∈ A d ( x, y ).If ( X, d ) is a metric space, then we denote by C bbs ( X ) the family of all bounded, continuousfunctions f : X → R having bounded support. Recall that the support of f is defined as theclosure of the set (cid:8) x ∈ X : f ( x ) = 0 (cid:9) . Moreover, by C b ( X ) we denote the family of allreal-valued, bounded continuous functions defined on X .3.2. Measure theory.
Let X be a non-empty set. Then a family A ⊂ X is said to be an algebra of sets provided X ∈ A and A \ B ∈ A for every A, B ∈ A . If in addition S n ∈ N A n ∈ A for any sequence ( A n ) n ∈ N ⊂ A , then A is called a σ -algebra . Given a family F ⊂ X , wedenote by σ ( F ) the σ -algebra generated by F , which is the smallest σ -algebra containing F .By measurable space we mean a couple ( X, A ), where X = ∅ is a set and A ⊂ X a σ -algebra.If ( X, A ) is a measurable space, Y a set, and ϕ : X → Y an arbitrary map, then we definethe pushforward σ -algebra of A under ϕ as ϕ ∗ A := (cid:8) B ⊂ Y (cid:12)(cid:12) ϕ − ( B ) ∈ A (cid:9) . Given two measurable spaces ( X, A X ), ( Y, A Y ) and a map ϕ : X → Y , we say that ϕ is measurable provided it holds that A Y ⊂ ϕ ∗ A X .A given function µ : A → [0 , + ∞ ], where A is an algebra on some non-empty set X , iscalled a set-function on ( X, A ) provided µ ( ∅ ) = 0. We say that a set-function µ on ( X, A ) isa finitely-additive measure provided it satisfies the following property: µ ( A ∪ B ) = µ ( A ) + µ ( B ) , for every A, B ∈ A such that A ∩ B = ∅ . If in addition A is a σ -algebra and the set-function µ satisfies µ (cid:0)S n ∈ N A n (cid:1) = P n ∈ N µ ( A n ) , whenever ( A n ) n ∈ N ⊂ A are pairwise disjoint , then we say that µ is a (countably-additive) measure on ( X, A ). By measure space we meana triple ( X, A , µ ), where ( X, A ) is a measurable space and µ is a measure on ( X, A ). We saythat µ is finite provided µ ( X ) < + ∞ , while it is σ -finite provided there exists a sequence( A n ) n ∈ N ⊂ A of measurable sets satisfying X = S n ∈ N A n and µ ( A n ) < + ∞ for every n ∈ N .Moreover, we say that the measure µ is concentrated on a set A ∈ A provided µ ( X \ A ) = 0.Given a non-empty set A ∈ A and a σ -algebra A ′ on A such that A ′ ⊂ A , we can considerthe restriction µ | A ′ of µ to A ′ . It trivially holds that µ | A ′ is a measure on ( A, A ′ ). In thecase where A ′ = A x A := (cid:8) A ′ ∈ A : A ′ ⊂ A (cid:9) , we just write µ x A in place of µ | A x A .If ( X, A X , µ ) is a measure space, ( Y, A Y ) a measurable space, and ϕ : X → Y a measurablemap, then we define the pushforward measure of µ under ϕ as( ϕ ∗ µ )( B ) := µ (cid:0) ϕ − ( B ) (cid:1) , for every B ∈ A Y . Observe that ϕ ∗ µ is a measure on ( Y, A Y ). If µ is finite, then ϕ ∗ µ is finite as well. On theother hand, if µ is σ -finite, then ϕ ∗ µ needs not be σ -finite, as easy counterexamples show. Ball measures and Borel measures.
Let (
X, d ) be a pseudometric space. Then wedenote by Ball( X ) the ball σ -algebra on X , which is defined as the σ -algebra generated bythe open balls (or, equivalently, by the closed balls). A measure on (cid:0) X, Ball( X ) (cid:1) is said to bea ball measure . Moreover, the Borel σ -algebra on X is the σ -algebra B ( X ) generated by thetopology of ( X, d ). A measure on (cid:0) X, B ( X ) (cid:1) is called a Borel measure . With a slight abuseof notation, we shall sometimes say that a measure is a ball measure (resp. a Borel measure)if it is defined on a σ -algebra containing the ball σ -algebra (resp. the Borel σ -algebra).A ball measure µ is said to be boundedly finite provided µ (cid:0) B ( x, r ) (cid:1) < + ∞ for every x ∈ X and r >
0. Observe that boundedly finite ball measures are σ -finite, as a consequence of theidentity X = S n ∈ N B (¯ x, n ), where ¯ x ∈ X is given. Since open balls are open sets, we haveBall( X ) ⊂ B ( X ) , for every pseudometric space ( X, d ) . However, it might happen (unless (
X, d ) is separable) that Ball( X ) = B ( X ). Definition 3.1 (Metric measure space) . By metric measure space we mean a triple ( X, d, m ) ,where ( X, d ) is a metric space, while m is a ball measure on X . Given a ball measure µ on a pseudometric space ( X, d ), we define its support spt( µ ) asspt( µ ) := n x ∈ X (cid:12)(cid:12)(cid:12) µ (cid:0) B ( x, r ) (cid:1) > r > o = X \ [ n B ( x, r ) (cid:12)(cid:12)(cid:12) x ∈ X, r > , µ (cid:0) B ( x, r ) (cid:1) = 0 o . A warning about our notation: if µ is a Borel measure (and ( X, d ) is not separable), then theabove definition of spt( µ ) might differ from other ones that appear in the literature. Let usnow collect in the following result a few basic properties of the support of a ball measure. Lemma 3.2.
Let ( X, d ) be a pseudometric space. Let µ be a ball measure on X . Then spt( µ ) is a closed set. If µ is σ -finite, then spt( µ ) is separable and in particular spt( µ ) ∈ Ball( X ) .Proof. First of all, observe that X \ spt( µ ) is open (as a union of open balls), thus spt( µ ) isclosed. Now assume that µ is σ -finite. We aim to show that spt( µ ) is separable. We argue bycontradiction: suppose spt( µ ) is not separable. Then we claim that there exist an uncountableset { x i } i ∈ I ⊂ spt( µ ) and ε > such that B ( x i , ε ) ∩ B ( x j , ε ) = ∅ for all i, j ∈ I with i = j . To prove it, for any n ∈ N pick a maximal n -separated subset S n of spt( µ ), whose existencefollows by a standard application of Zorn’s lemma. Namely, we have that d ( x, y ) ≥ n , for every x, y ∈ S n with x = y, ∀ z ∈ spt( µ ) ∃ x ∈ S n : d ( x, z ) < n . Then S n ∈ N S n is dense in spt( µ ) by construction. Being spt( µ ) non-separable, we deducethat S ¯ n must be uncountable for some ¯ n ∈ N . Call { x i } i ∈ I := S ¯ n and ε := 1 / ¯ n . Observe that B ( x i , ε ) ∩ B ( x j , ε ) = ∅ , for every i, j ∈ I with i = j, (3.1) LTRALIMITS OF POINTED METRIC MEASURE SPACES 13 thus the claim is proven. Given that µ is σ -finite, we can find a sequence ( A n ) n ∈ N ⊂ Ball( X )such that X = S n ∈ N A n and µ ( A n ) < + ∞ for all n ∈ N . Given any i ∈ I , there exists n i ∈ N such that µ (cid:0) A n i ∩ B ( x i , ε ) (cid:1) >
0. Let us now define I nk := n i ∈ I (cid:12)(cid:12)(cid:12) n i = n, µ (cid:0) A n ∩ B ( x i , ε ) (cid:1) ≥ /k o , for every n, k ∈ N . Given any finite subset F of I nk , we may estimate Fk ≤ X i ∈ F µ (cid:0) A n ∩ B ( x i , ε ) (cid:1) (3.1) = µ (cid:0) A n ∩ S i ∈ F B ( x i , ε ) (cid:1) ≤ µ ( A n ) , whence it follows that I nk ≤ k µ ( A n ) and in particular I nk is finite. This is in contradictionwith the fact that I = S n,k ∈ N I nk is uncountable. Therefore, spt( µ ) is proven to be separable.Finally, by using both the closedness and the separability of spt( µ ), we see thatspt( µ ) = \ k ∈ N [ n ∈ N B ( x n , /k ) , where ( x n ) n ∈ N ⊂ spt( µ ) is any dense sequence. This yields spt( µ ) ∈ Ball( X ), as required. (cid:3) Remark 3.3.
In the last part of the proof of Lemma 3.2, we used the following general fact:
Let ( X, d ) be a pseudometric space. Then C ∈ Ball( X ) for every C ⊂ X closed and separable. Indeed, given any dense sequence ( x n ) n ∈ N in C , it holds C = T k ∈ N S n ∈ N B ( x n , /k ). (cid:4) Pointed measured Gromov–Hausdorff convergence
With a slight abuse of notation, we say that any complete and separable metric space (
X, d )is a
Polish metric space . (Typically, this terminology is referring to a topological space whosetopology is induced by a complete and separable distance.)
Definition 4.1 (( R, ε )-approximation) . Let ( X, d X , p X ) , ( Y, d Y , p Y ) be pointed Polish metricspaces. Let R, ε > be such that ε < R . Then a given Borel map ψ : B ( p X , R ) → Y is saidto be a ( R, ε )-approximation provided it satisfies ψ ( p X ) = p Y , sup x,y ∈ B ( p X ,R ) (cid:12)(cid:12)(cid:12) d X ( x, y ) − d Y (cid:0) ψ ( x ) , ψ ( y ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ ε, B ( p Y , R − ε ) ⊂ ψ (cid:0) B ( p X , R ) (cid:1) ε . Remark 4.2. (1) Notice that it is always possible to extend the map (in a Borel manner) to the wholespace X . Therefore, we will often consider ( R, ε )-approximations as maps ψ : X → Y to avoid unnecessary complication in the presentation.(2) Being ( R, ε )-approximation means – roughly speaking – that ψ is roughly isometricand roughly surjective to the corresponding ball on the image side. (cid:4) Lemma 4.3.
Let ( X, d X , p X ) , ( Y, d Y , p Y ) be pointed Polish metric spaces. Let ψ : X → Y be a given ( R, ε ) -approximation, for some R, ε > satisfying ε < R . Then there exists a ( R − ε, ε ) -approximation φ : Y → X – that we will call a quasi-inverse of ψ – such that d X (cid:0) x, ( φ ◦ ψ )( x ) (cid:1) < ε, for every x ∈ B ( p X , R − ε ) , (4.1a) d Y (cid:0) y, ( ψ ◦ φ )( y ) (cid:1) < ε, for every y ∈ B ( p Y , R − ε ) . (4.1b) Proof.
Let ( x i ) i be a dense sequence in B ( p X , R ). Given any y ∈ B ( p Y , R − ε ) \ { p Y } , wedefine φ ( y ) := x i y , where i y is the smallest index i ∈ N for which d Y (cid:0) ψ ( x i ) , y (cid:1) < ε . Also, weset φ ( y ) := ˜ x for all y / ∈ B ( p Y , R − ε ) \ { p Y } , where ˜ x is any given element of X \ B ( p X , R − ε )if X \ B ( p X , R − ε ) = ∅ and ˜ x := p X otherwise. The resulting map φ : Y → X is Borel andsatisfies φ ( p Y ) = p X by construction. For every y, y ′ ∈ B ( p Y , R − ε ), we have that (cid:12)(cid:12)(cid:12) d X (cid:0) φ ( y ) , φ ( y ′ ) (cid:1) − d Y ( y, y ′ ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) d X (cid:0) φ ( y ) , φ ( y ′ ) (cid:1) − d Y (cid:0) ( ψ ◦ φ )( y ) , ( ψ ◦ φ )( y ′ ) (cid:1)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) d Y (cid:0) ( ψ ◦ φ )( y ) , ( ψ ◦ φ )( y ′ ) (cid:1) − d Y ( y, y ′ ) (cid:12)(cid:12)(cid:12) ≤ ε + d Y (cid:0) ( ψ ◦ φ )( y ) , y (cid:1) + d Y (cid:0) ( ψ ◦ φ )( y ′ ) , y ′ (cid:1) < ε, which shows that sup y,y ′ ∈ B ( p Y ,R − ε ) (cid:12)(cid:12) d X (cid:0) φ ( y ) , φ ( y ′ ) (cid:1) − d Y ( y, y ′ ) (cid:12)(cid:12) ≤ ε . Moreover, given any x ∈ B ( p X , R − ε ), we have that ψ ( x ) ∈ B ( p Y , R − ε ), thus from the validity of the inequalities d X (cid:0) ( φ ◦ ψ )( x ) , x (cid:1) ≤ d Y (cid:0) ( ψ ◦ φ ◦ ψ )( x ) , ψ ( x ) (cid:1) + ε < ε < ε (4.2)it follows that x belongs to the 3 ε -neighbourhood of φ (cid:0) B ( p Y , R − ε ) (cid:1) . In particular, it holdsthat B ( p X , R − ε ) ⊂ φ (cid:0) B ( p Y , R − ε ) (cid:1) ε , whence accordingly φ is a ( R − ε, ε )-approximation.In order to conclude, it only remains to observe that (4.1b) is a direct consequence of thevery definition of φ , while (4.1a) follows from the estimate in (4.2). (cid:3) Lemma 4.4.
Let ( X, d X , p X ) , ( Y, d Y , p Y ) be pointed Polish metric spaces. Let R, r, r ′ , ε > be such that r + r ′ < R − ε and r > ε . Let ψ : X → Y be a given ( R, ε ) -approximation, withquasi-inverse φ : Y → X . Then for every point y ∈ B ( p Y , r ′ ) it holds that B ( y, r − ε ) ⊂ ψ (cid:0) B ( φ ( y ) , r ) (cid:1) ε , (4.3a) ψ − (cid:0) B ( y, r − ε ) (cid:1) ⊂ B (cid:0) φ ( y ) , r + 4 ε (cid:1) . (4.3b) Proof.
Let y ∈ B ( p Y , r ′ ) be fixed. Pick any point z ∈ B ( y, r − ε ). We aim to show that wehave z ∈ ψ (cid:0) B ( φ ( y ) , r ) (cid:1) ε , whence (4.3a) would follow. Notice that d Y ( y, p Y ) < r ′ < R − ε and d Y ( z, p Y ) ≤ d Y ( z, y ) + d Y ( y, p Y ) < ( r − ε ) + r ′ < R − ε , thus y, z ∈ B ( p Y , R − ε ) and d X (cid:0) φ ( z ) , φ ( y ) (cid:1) ≤ d Y ( z, y ) + 3 ε < ( r − ε ) + 3 ε = r, where we used the fact that φ is a ( R − ε, ε )-approximation. Then φ ( z ) ∈ B (cid:0) φ ( x ) , r (cid:1) , sothat from the estimate d Y (cid:0) z, ( ψ ◦ φ )( z ) (cid:1) < ε – which is granted by (4.1b) – we deduce thatthe point z belongs to ψ (cid:0) B ( φ ( y ) , r ) (cid:1) ε , as required.In order to prove (4.3b), we argue by contradiction: suppose there is z ∈ X \ B (cid:0) φ ( y ) , r +4 ε (cid:1) with ψ ( z ) ∈ B ( y, r − ε ). Since B ( y, r − ε ) ⊂ B ( p Y , R ) and ψ (cid:0) X \ B ( p X , R ) (cid:1) ⊂ Y \ B ( p Y , R ), LTRALIMITS OF POINTED METRIC MEASURE SPACES 15 we have that z ∈ B ( p X , R ). Moreover, (4.3a) grants the existence of w ∈ B (cid:0) φ ( y ) , r (cid:1) such that d Y (cid:0) ψ ( z ) , ψ ( w ) (cid:1) < ε . Notice that w ∈ B ( p X , R ), as it is granted by the following estimates: d X ( w, p X ) ≤ d X (cid:0) w, φ ( y ) (cid:1) + d X (cid:0) φ ( y ) , p X (cid:1) < r + d X (cid:0) φ ( y ) , φ ( p Y ) (cid:1) ≤ r + d Y ( y, p Y ) + 3 ε < r + r ′ + 3 ε < R. Therefore, we deduce that d X ( z, w ) ≤ d Y (cid:0) ψ ( z ) , ψ ( w ) (cid:1) + ε < ε . On the other hand, it holds d X ( z, w ) ≥ d X (cid:0) z, φ ( y ) (cid:1) − d X (cid:0) w, φ ( y ) (cid:1) > ( r + 4 ε ) − r = 4 ε, which leads to a contradiction. Consequently, the claimed inclusion (4.3b) is proven. (cid:3) Definition 4.5 (Pointed Gromov–Hausdorff convergence) . Let (cid:8) ( X i , d i , p i ) (cid:9) i ∈ ¯ N be a sequenceof pointed Polish metric spaces. Then we say that ( X i , d i , p i ) converges to ( X ∞ , d ∞ , p ∞ ) inthe pointed Gromov–Hausdorff sense (briefly, pGH sense ) as i → ∞ provided there exists asequence ( ψ i ) i ∈ N of ( R i , ε i ) -approximations ψ i : X i → X ∞ , for some R i ր + ∞ and ε i ց . Definition 4.6 (Pointed measured Gromov–Hausdorff convergence) . Let (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ ¯ N be a sequence of pointed Polish metric measure spaces, with m i boundedly finite. Then wesay that the sequence ( X i , d i , m i , p i ) converges to ( X ∞ , d ∞ , m ∞ , p ∞ ) in the pointed measuredGromov–Hausdorff sense (briefly, pmGH sense ) provided ( X i , d i , p i ) → ( X ∞ , d ∞ , p ∞ ) in thepGH sense and there exists a sequence ( ψ i ) i ∈ N of ( R i , ε i ) -approximations exploiting the pGHconvergence such that ( ψ i ) ∗ m i ⇀ m ∞ in duality with C bbs ( X ∞ ) , namely, ˆ ϕ ◦ ψ i d m i −→ ˆ ϕ d m ∞ , for every ϕ ∈ C bbs ( X ∞ ) . Let us illustrate which is the relation between pGH convergence and pmGH convergence.On the one hand, given a sequence (cid:0) ( X i , d i , m i , p i ) (cid:1) that converges to ( X ∞ , d ∞ , m ∞ , p ∞ ) in thepmGH sense, it holds that (cid:0) ( X i , d i , p i ) (cid:1) converges to ( X ∞ , d ∞ , p ∞ ) in the pGH sense. On theother hand, if we consider a sequence (cid:0) ( X i , d i , p i ) (cid:1) that pGH-converges to ( X ∞ , d ∞ , p ∞ ) anda measure m ∞ on X ∞ , we can find a sequence of measures ( m i ) i ∈ N such that (cid:0) ( X i , d i , m i , p i ) (cid:1) pmGH-converges to ( X ∞ , d ∞ , m ∞ , p ∞ ); the latter claim is the content of the following result.This shows that – in a sense – the pmGH convergence is essentially a metric concept, wherea control on the behaviour of the measures is added subsequently. Lemma 4.7.
Let (cid:8) ( X i , d i , p i ) (cid:9) i ∈ ¯ N be a sequence of pointed Polish metric spaces. Suppose that ( X i , d i , p i ) → ( X ∞ , d ∞ , p ∞ ) in the pGH sense. Let ψ i : X i → X ∞ be ( R i , ε i ) -approximationsand m ∞ a Borel probability measure on X ∞ . Then there exists a sequence ( m i ) i ∈ N of Borelprobability measures m i ∈ P ( X i ) such that ( ψ i ) ∗ m i ⇀ m ∞ , in duality with C bbs ( X ∞ ) . (4.4) In particular, it holds that ( X i , d i , m i , p i ) → ( X ∞ , d ∞ , m ∞ , p ∞ ) in the pmGH sense.Proof. Up to a diagonalisation argument, it is sufficient to prove the claim for a fully-supportedmeasure m ∞ of the form m ∞ := X j ∈ N λ j δ x j , (4.5) where ( x j ) j ∈ N is dense in X ∞ and ( λ j ) j ∈ N ⊂ [0 ,
1] satisfies P j ∈ N λ j = 1. Indeed, thesemeasures are weakly dense in the space P ( X ∞ ). Then let m ∞ be a fixed measure as in (4.5).Given any i ∈ N and x ∈ B ( p i , R i ), we define j ( i, x ) ∈ N as the smallest number j ∈ N suchthat d ∞ (cid:0) ψ i ( x ) , x j (cid:1) < ε i . Let us define J i := (cid:8) j ( i, x ) : x ∈ B ( p i , R i ) (cid:9) ⊂ N for every i ∈ N .Given any j ∈ J i , choose any point x ij ∈ B ( p i , R i ) such that j ( i, x ij ) = j . Observe that d ∞ (cid:0) ψ i ( x ij ) , x j (cid:1) < ε i , for every i ∈ N and j ∈ J i . (4.6)Now for any i ∈ N let us define the Borel probability measure m i on X i as m i := 1 c i X j ∈ J i λ j δ x ij , where we set c i := X j ∈ J i λ j . We claim that ∀ k ∈ N ∃ ¯ i ∈ N : ∀ i ≥ ¯ i { , . . . , k } ⊂ J i . (4.7)In order to prove it, let k ∈ N be fixed. Choose any ¯ i ∈ N so that x , . . . , x k ∈ B ( p ∞ , R i − ε i ) , for every i ≥ ¯ i,d ∞ ( x j , x j ′ ) > ε i , for every i ≥ ¯ i and j, j ′ ∈ { , . . . , k } with j = j ′ . (4.8)Given any j ∈ { , . . . , k } , it follows from the first line in (4.8) that for every i ≥ ¯ i there existsa point x ∈ B ( p i , R i ) such that d ∞ (cid:0) ψ i ( x ) , x j (cid:1) < ε i . Since the second line in (4.8) yields d ∞ (cid:0) ψ i ( x ) , x j ′ (cid:1) ≥ d ∞ ( x j , x j ′ ) − d ∞ (cid:0) ψ i ( x ) , x j (cid:1) > ε i − ε i = ε i , for every j ′ < j, we deduce that j = j ( i, x ) and accordingly j ∈ J i . This proves the validity of the claim (4.7).It remains to prove that ( ψ i ) ∗ m i ⇀ m ∞ in duality with C bbs ( X ∞ ). Let f ∈ C bbs ( X ∞ ) befixed and put M := sup X ∞ | f | . Fix any ε >
0. Pick any k ∈ N such that M P j>k λ j < ε . Byusing (4.7) we can find ¯ i ∈ N such that { , . . . , k } ⊂ J i for every i ≥ ¯ i . By using (4.6) andthe continuity of the function f , we see thatlim i →∞ f (cid:0) ψ i ( x ij ) (cid:1) = f ( x j ) , for every j ∈ { , . . . , k } . (4.9)Moreover, for every i ≥ ¯ i we can estimate (cid:12)(cid:12)(cid:12)(cid:12) c i ˆ f d( ψ i ) ∗ m i − ˆ f d m ∞ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) X j ∈ J i λ j f (cid:0) ψ i ( x ij ) (cid:1) − X j ∈ N λ j f ( x j ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 λ j (cid:0) f (cid:0) ψ i ( x ij ) (cid:1) − f ( x j ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) + X j ∈ J i : j>k λ j (cid:12)(cid:12) f (cid:0) ψ i ( x ij ) (cid:1)(cid:12)(cid:12) + X j>k λ j (cid:12)(cid:12) f ( x j ) (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 λ j (cid:0) f (cid:0) ψ i ( x ij ) (cid:1) − f ( x j ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) + 2 M X j>k λ j ≤ (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 λ j (cid:0) f (cid:0) ψ i ( x ij ) (cid:1) − f ( x j ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) + 2 ε. LTRALIMITS OF POINTED METRIC MEASURE SPACES 17
By letting i → ∞ and using (4.9), we deduce that lim i (cid:12)(cid:12) c i ´ f d( ψ i ) ∗ m i − ´ f d m ∞ (cid:12)(cid:12) ≤ ε ,whence by arbitrariness of ε > i c i ´ f d( ψ i ) ∗ m i = ´ f d m ∞ . Moreover, itfollows from (4.7) that lim i c i = 1, so that lim i ´ f d( ψ i ) ∗ m i = ´ f d m ∞ for all f ∈ C bbs ( X ∞ ),thus proving the validity of (4.4). (cid:3) Ultrafilters and ultraproducts
The notion of ultralimit of pointed metric measure spaces introduced in this paper, aswell as the well-known notion of ultralimit of pointed metric spaces, rely on the concepts ofultrafilter and ultralimits in metric spaces. We recall the necessary definitions and fix someterminology.
Definition 5.1 (Ultrafilter) . Let ω ⊂ N be a collection of subsets of natural numbers. Theset ω is an ultrafilter on N if the following three conditions hold: (1) If A, B ∈ ω , then A ∩ B ∈ ω . (2) If A ∈ ω and A ⊂ B ⊂ N , then B ∈ ω . (3) If A ⊂ N , then either A ∈ ω or N \ A ∈ ω .An ultrafilter ω is called non-principal if it does not contain any singleton. Remark 5.2.
For a set ω ⊂ N , being an ultrafilter is equivalent to the characteristic function χ ω of ω being a non-trivial, finitely-additive measure on 2 N . Due to this correspondence, wewill use the phrasing “for ω - almost every” to mean “outside of a χ ω - null set”. (cid:4) Definition 5.3 (Ultralimits in metric spaces) . Let ω ⊂ N be a given non-principal ultrafilter.Let ( X, d ) be a metric space and ( x i ) a sequence of points in X . Then a point x ∞ ∈ X is saidto be the ultralimit , or the ω -limit , of the sequence ( x i ) if for every ε > we have that (cid:8) i ∈ N (cid:12)(cid:12) d ( x ∞ , x i ) ≤ ε (cid:9) ∈ ω. In this case, we write x ∞ = lim i → ω x i . We will use the following simple argument several times.
Lemma 5.4.
For every i ∈ N , let F i ∈ ω be so that F i ⊃ F i +1 and F i ⊂ { k ≥ i } . Then forevery k ∈ F i , there exists i k ∈ N so that k ∈ F i k \ F i k +1 . Moreover, it holds that lim k → ω i k = ∞ . Proof.
The existence of i k is clearly true. To show that lim k → ω i k = ∞ , it is enough to noticethat { k ∈ N : i k ≥ n } ⊃ F n ∈ ω. (cid:3) For most of the paper, ω will be an arbitrary non-principal ultrafilter on N . Only at the endof Appendix A, we consider ultrafilters having the property described in the ensuing result: Lemma 5.5.
Let { N n } n ∈ N be a partition of N such that N n is an infinite set for all n ∈ N .Then there exists a non-principal ultrafilter ω on N that contains all those subsets S ⊂ N suchthat N n \ S is finite for all but finitely many n ∈ N . In particular, N \ N n ∈ ω for every n ∈ N . Proof.
Let β be the family of all those subsets S ⊂ N such that N n \ S is finite for all butfinitely many n ∈ N . We claim that β is a filter on N , i.e. , it satisfies (1), (2) of Definition 5.1and does not contain ∅ . The latter is trivially verified, as N n \ ∅ is infinite for all n ∈ N andthus ∅ / ∈ β . Given any S, T ∈ β , we have that N n \ ( A ∩ B ) = ( N n \ A ) ∪ ( N n \ B ) is finitefor all but finitely many n , so that A ∩ B ∈ β . Moreover, if S ∈ β and T ⊂ N satisfy S ⊂ T ,then the set N n \ T ⊂ N n \ S is finite for all but finitely many n , which shows that T ∈ β as well. All in all, we have proven that β is a filter on N . An application of the UltrafilterLemma yields the existence of an ultrafilter ω on N such that β ⊂ ω . Since N m \ ( N \ N n ) = ∅ for every n, m ∈ N with n = m , we see that { N \ N n } n ∈ N ⊂ β ⊂ ω . This implies that ω isnon-principal, as T S ∈ ω S ⊂ T n ∈ N N \ N n = ∅ . Consequently, the statement is achieved. (cid:3) Ultralimits of metric spaces
In this section, we will recall the definition of ultralimit of (pointed) metric spaces. As anintermediate step, we will use the ultraproduct of sets – or rather the set-theoretic ultralimit– in which a pseudometric is defined in a natural manner. The ultralimit of a sequence ofmetric spaces is then defined by identifying those points which are at zero distance from eachother. Defining the ultralimit of pointed metric spaces in such a way is for introducing thenecessary terminology for the definition of ultralimits of metric measure spaces in ChapterIII.Let ω ⊂ N be a non-principal ultrafilter on N . Actually, ω will remain fixed throughout the whole paper, with the only exception of Proposition A.6. Let (cid:0) ( X i , d i ) (cid:1) be a sequence of metric spaces.Let us define the family Π ωi ∈ N X i as Π ωi ∈ N X i := G S ∈ ω Y i ∈ S X i . Given any x ∈ Π ωi ∈ N X i , we will denote by S x the element of ω for which x ∈ Q i ∈ S x X i .However, often we will just write x = ( x i ), omitting the explicit reference to the index set S x .Consider the following equivalence relation on Π ωi ∈ N X i : given ( x i ) , ( y i ) ∈ Π ωi ∈ N X i , we declare( x i ) ∼ ω ( y i ) ⇐⇒ (cid:8) j ∈ S ( x i ) ∩ S ( y i ) (cid:12)(cid:12) x j = y j (cid:9) ∈ ω. Then we define the ultraproduct ¯ X ω of the sets X i as the quotient set¯ X ω := Π ωi ∈ N X i (cid:14) ∼ ω . The equivalence class of an element ( x i ) ∈ Π ωi ∈ N X i with respect to ∼ ω will be denoted by[ x i ] ∈ ¯ X ω , while by π ω : Π ωi ∈ N X i → ¯ X ω we mean the canonical projection map π ω (cid:0) ( x i ) (cid:1) := [ x i ]. Definition 6.1 (Set-theoretic ultralimit of metric spaces) . Let (cid:0) ( X i , d i ) (cid:1) be a sequence ofmetric spaces. Then we define its set-theoretic ultralimit as Π i → ω ( X i , d i ) := ( ¯ X ω , ¯ d ω ) , LTRALIMITS OF POINTED METRIC MEASURE SPACES 19 where ¯ d ω is the extended pseudodistance on the ultraproduct ¯ X ω , which is defined by ¯ d ω (cid:0) [ x i ] , [ y i ] (cid:1) := lim i → ω d i ( x i , y i ) ∈ [0 , + ∞ ] , for every [ x i ] , [ y i ] ∈ ¯ X ω . (6.1)The definition in (6.1) is well-posed, as the quantity lim i → ω d i ( x i , y i ) does not depend onthe specific choice of the representatives of [ x i ] and [ y i ]. Definition 6.2 (Set-theoretic ultralimit of pointed metric spaces) . Let (cid:0) ( X i , d i , p i ) (cid:1) be asequence of pointed metric spaces. Then we define its set-theoretic ultralimit as Π i → ω ( X i , d i , p i ) := (cid:0) O ( ¯ X ω ) , ¯ d ω , [ p i ] (cid:1) , where we set O ( ¯ X ω ) := n [ x i ] ∈ ¯ X ω (cid:12)(cid:12)(cid:12) ¯ d ω (cid:0) [ x i ] , [ p i ] (cid:1) < + ∞ o . The set-theoretic ultralimit of pointed metric spaces is a (pointed) pseudometric space.Now the ultralimit can be defined as the natural corresponding metric space.
Definition 6.3 (Ultralimit of pointed metric spaces) . Let (cid:0) ( X i , d i , p i ) (cid:1) be a sequence ofpointed metric spaces. Let us consider the following equivalence relation on O ( ¯ X ω ) : givenany [ x i ] , [ y i ] ∈ O ( ¯ X ω ) , we declare that [ x i ] ∼ [ y i ] if and only if ¯ d ω (cid:0) [ x i ] , [ y i ] (cid:1) = 0 . We denoteby [[ x i ]] the equivalence class of [ x i ] . Then we define the ultralimit of (cid:0) ( X i , d i , p i ) (cid:1) as lim i → ω ( X i , d i , p i ) := ( X ω , d ω , p ω ) , where we set X ω := O ( ¯ X ω ) / ∼ , d ω (cid:0) [[ x i ]] , [[ y i ]] (cid:1) := ¯ d ω (cid:0) [ x i ] , [ y i ] (cid:1) , for every [[ x i ]] , [[ y i ]] ∈ X ω , and p ω := [[ p i ]] . Observe that the value of d ω (cid:0) [[ x i ]] , [[ y i ]] (cid:1) does not depend on the chosen representatives of[[ x i ]] and [[ y i ]]. The ultralimit lim i → ω ( X i , d i , p i ) is a pointed metric space. We denote by π (cid:0) [ x i ] (cid:1) := [[ x i ]] , for every [ x i ] ∈ O ( ¯ X ω ) , (6.2)the natural projection map π : O ( ¯ X ω ) → X ω on the quotient.HAPTER III Ultralimits of metric measure spaces
In this chapter, we will introduce the notion of ultralimit of pointed metric measure spaces,which naturally arises from the related notions. Yet, we are not aware of any usage of it inthe literature. 7.
Construction of the ultralimit
We begin by introducing a relevant family of subsets of the set-theoretic ultralimit of metricspaces, which will play a key role in the sequel.
Definition 7.1 (Internal measurable set) . Let (cid:0) ( X i , d i ) (cid:1) be a sequence of metric spaces.Then a set ¯ A ⊂ ¯ X ω is said to be an internal measurable set provided there exist S ∈ ω and ( A i ) i ∈ S ∈ Q i ∈ S Ball( X i ) with ¯ A = π ω (cid:0) Q i ∈ S A i (cid:1) . The family of all internal measurablesubsets of ¯ X ω is denoted by A ( ¯ X ω ) . Notice that ¯ A coincides with the ultraproduct of the sets A i , which can be seen as a subsetof ¯ X ω in a canonical way. In view of this observation, we will denote it by ¯ A = Π i → ω A i . Lemma 7.2.
Let (cid:0) ( X i , d i ) (cid:1) be a sequence of metric spaces. Let ( A i ) i ∈ S ∈ Q i ∈ S Ball( X i ) and ( B i ) i ∈ S ′ ∈ Q i ∈ S ′ Ball( X i ) be given, for some S, S ′ ∈ ω . Then it holds that Π i → ω A i = Π i → ω B i ⇐⇒ A i = B i for ω -a.e. i ∈ S ∩ S ′ . Proof.
Suppose A i = B i holds for ω -a.e. i ∈ S ∩ S ′ . Set T := { i ∈ S ∩ S ′ : A i = B i } ∈ ω . If[ x i ] ∈ Π i → ω A i , then T ∩{ j ∈ S ∩ S ( x i ) : x j ∈ A j } ∈ ω , so its superset { j ∈ S ′ ∩ S ( x i ) : x j ∈ B j } belongs to ω as well, which shows that [ x i ] ∈ Π i → ω B i and thus Π i → ω A i ⊂ Π i → ω B i . Theconverse inclusion follows by an analogous argument, just interchanging ( A i ) i ∈ S and ( B i ) i ∈ S ′ .On the contrary, suppose { i ∈ S ∩ S ′ : A i = B i } ∈ ω . Possibly interchanging ( A i ) i ∈ S and( B i ) i ∈ S ′ , we can suppose that T := { i ∈ S ∩ S ′ : A i \ B i = ∅} ∈ ω . Pick any x i ∈ A i \ B i forevery i ∈ T . Then it holds [ x i ] ∈ Π i → ω A i \ Π i → ω B i , thus proving that Π i → ω A i = Π i → ω B i . (cid:3) The idea is to define the limit measure on a suitable algebra A O ( ¯ X ω ) (obtained by re-striction from A ( ¯ X ω )) of subsets of O ( ¯ X ω ) using the measure on internal measurable subsetsdefined in a natural way, and then extend it to a σ -algebra containing them. Finally, themeasure on the ultralimit of pointed metric measure spaces is obtained by pushforward underthe projection map π . We will use the following proposition. Proposition 7.3.
Let (cid:0) ( X i , d i ) (cid:1) be a sequence of metric spaces. Then it holds that (Π i → ω A i ) ∪ (Π i → ω B i ) = Π i → ω A i ∪ B i , (7.1a)(Π i → ω A i ) c = Π i → ω A ci , (7.1b) for every Π i → ω A i , Π i → ω B i ∈ A ( ¯ X ω ) . In particular, the family A ( ¯ X ω ) is an algebra of subsetsof ¯ X ω . Moreover, if m i is a given ball measure on the space X i for every i ∈ N , then theset-function ˜ m ω : A ( ¯ X ω ) → [0 , + ∞ ] defined by ˜ m ω (Π i → ω A i ) := lim i → ω m i ( A i ) , for every Π i → ω A i ∈ A ( ¯ X ω ) is a finitely-additive measure on A ( ¯ X ω ) . The set-function ˜ m ω is well-defined due to Lemma7.2.Proof. In order to prove (7.1a), first observe that, trivially, Π i → ω A i , Π i → ω B i ⊂ Π i → ω A i ∪ B i .Conversely, let [ x i ] ∈ Π i → ω A i ∪ B i be fixed. Pick any S ⊂ S ( x i ) such that x j ∈ A j for all j ∈ S and x j ∈ B j for all j ∈ S ( x i ) \ S . Then either S ∈ ω and so [ x i ] ∈ Π i → ω A i , or S ( x i ) \ S ∈ ω and so [ x i ] ∈ Π i → ω B i . We have proven Π i → ω A i ∪ B i ⊂ (Π i → ω A i ) ∪ (Π i → ω B i ), yielding (7.1a).In order to prove (7.1b), observe that for any given [ x i ] ∈ ¯ X ω we have that[ x i ] ∈ Π i → ω A i ⇐⇒ { j ∈ S ( x i ) : x j ∈ A j } ∈ ω ⇐⇒ { j ∈ S ( x i ) : x j ∈ A cj } / ∈ ω ⇐⇒ [ x i ] ∈ (Π i → ω A ci ) c , proving (7.1b). Given that also ∅ = Π i → ω ∅ ∈ A ( ¯ X ω ), we thus see that A ( ¯ X ω ) is an algebra.Let us now consider the set-function ˜ m ω . Fix pairwise disjoint sets ¯ A , . . . , ¯ A n ∈ A ( ¯ X ω ),say ¯ A j = Π i → ω A ji for all j = 1 , . . . , n . Given j, j ′ = 1 , . . . , n with j = j ′ , we infer from (7.1a)and (7.1b) that Π i → ω A ji ∩ A j ′ i = ¯ A j ∩ ¯ A j ′ = ∅ , whence Lemma 7.2 grants that A ji ∩ A j ′ i = ∅ for ω -a.e. i . Therefore, we conclude that˜ m ω ( ¯ A ∪ · · · ∪ ¯ A n ) = ˜ m ω (cid:0) Π i → ω A i ∪ · · · ∪ A ni (cid:1) = lim i → ω m i ( A i ∪ · · · ∪ A ni ) = lim i → ω n X j =1 m i ( A ji )= n X j =1 lim i → ω m i ( A ji ) = n X j =1 ˜ m ω ( ¯ A j ) , thus proving that ˜ m ω is finitely-additive. Consequently, the proof is achieved. (cid:3) Definition 7.4 (The algebra A O ( ¯ X ω )) . Let (cid:0) ( X i , d i , p i ) (cid:1) be a sequence of pointed metricmeasure spaces. Then we define the algebra A O ( ¯ X ω ) as A O ( ¯ X ω ) := (cid:8) ¯ A ∩ O ( ¯ X ω ) (cid:12)(cid:12) ¯ A ∈ A ( ¯ X ω ) (cid:9) . In general, the set O ( ¯ X ω ) is not an internal measurable subset of ¯ X ω . However, it is in any σ -algebra containing A ( ¯ X ω ) due to the following proposition, see also [18, Lemma 3.1]. Proposition 7.5.
Let (cid:0) ( X i , d i , p i ) (cid:1) be a sequence of pointed metric spaces. Then it holds that ¯ B (cid:0) [ x i ] , R (cid:1) = \ n ∈ N Π i → ω B ( x i , R + 1 /n ) , for every [ x i ] ∈ ¯ X ω and R > . (7.2) In particular, B ∈ σ (cid:0) A ( ¯ X ω ) (cid:1) for every closed ball B ⊂ ¯ X ω , thus O ( ¯ X ω ) ∈ σ (cid:0) A ( ¯ X ω ) (cid:1) as acountable union of closed balls. Moreover, for any set Π i → ω A i ∈ A ( ¯ X ω ) we have that (Π i → ω A i ) ∩ O ( ¯ X ω ) = [ R ∈ N Π i → ω (cid:0) A i ∩ B ( p i , R ) (cid:1) . (7.3) Proof.
First, let us prove (7.2). If [ y i ] ∈ ¯ B (cid:0) [ x i ] , R (cid:1) and n ∈ N , then lim i → ω d i ( x i , y i ) < R +1 /n ,which implies that (cid:8) j ∈ S ( x i ) ∩ S ( y i ) (cid:12)(cid:12) y j ∈ B ( x j , R + 1 /n ) (cid:9) = (cid:8) j ∈ S ( x i ) ∩ S ( y i ) (cid:12)(cid:12) d j ( x j , y j ) < R + 1 /n (cid:9) ∈ ω. Therefore, by definition of internal measurable set, we have that [ y i ] ∈ Π i → ω B ( x i , R + 1 /n ),thus ¯ B (cid:0) [ x i ] , R (cid:1) ⊂ T n ∈ N Π i → ω B ( x i , R + 1 /n ). Conversely, fix [ y i ] ∈ T n ∈ N Π i → ω B ( x i , R + 1 /n ).Given any ε >
0, pick some n ∈ N satisfying n > /ε . Then we have that ω ∋ (cid:8) j ∈ S ( x i ) ∩ S ( y i ) (cid:12)(cid:12) y j ∈ B ( x j , R + 1 /n ) (cid:9) = (cid:8) j ∈ S ( x i ) ∩ S ( y i ) (cid:12)(cid:12) d j ( x j , y j ) < R + 1 /n (cid:9) ⊂ (cid:8) j ∈ S ( x i ) ∩ S ( y i ) (cid:12)(cid:12) d j ( x j , y j ) < R + ε (cid:9) , which yields (cid:8) j ∈ S ( x i ) ∩ S ( y i ) (cid:12)(cid:12) d j ( x j , y j ) < R + ε (cid:9) ∈ ω . This grants that ¯ d ω (cid:0) [ x i ] , [ y i ] (cid:1) ≤ R byarbitrariness of ε >
0, so that [ y i ] ∈ ¯ B (cid:0) [ x i ] , R (cid:1) and accordingly the identity in (7.2) is proven.Let us now show (7.3). Fix any [ x i ] ∈ (Π i → ω A i ) ∩ O ( ¯ X ω ). Given that [ x i ] ∈ O ( ¯ X ω ), thereexists R ′ ∈ N such that (cid:8) j ∈ S ( x i ) : d j ( x j , p j ) < R ′ (cid:9) ∈ ω , so that [ x i ] ∈ Π i → ω B ( p i , R ′ ). Thisensures that [ x i ] ∈ Π i → ω (cid:0) A i ∩ B ( p i , R ′ ) (cid:1) ⊂ S R ∈ N Π i → ω (cid:0) A i ∩ B ( p i , R ) (cid:1) by Proposition 7.3.Conversely, fix [ x i ] ∈ S R ∈ N Π i → ω (cid:0) A i ∩ B ( p i , R ) (cid:1) . Then there are S ∈ ω and R ∈ N such that { i ∈ S | x i ∈ A i } ∩ (cid:8) i ∈ S (cid:12)(cid:12) d i ( x i , p i ) < R (cid:9) = (cid:8) i ∈ S (cid:12)(cid:12) x i ∈ A i ∩ B ( p i , R ) (cid:9) ∈ ω. By exploiting the upward-closedness of the ultrafilter ω , we infer that { i ∈ S | x i ∈ A i } ∈ ω and (cid:8) i ∈ S (cid:12)(cid:12) d i ( x i , p i ) < R (cid:9) ∈ ω , which implies that [ x i ] ∈ Π i → ω A i and ¯ d ω (cid:0) [ x i ] , [ p i ] (cid:1) ≤ R ,respectively. In particular, it holds [ x i ] ∈ (Π i → ω A i ) ∩ O ( ¯ X ω ). All in all, (7.3) is proven. (cid:3) We may now give the definition of the finitely-additive measure ¯ m ω on the algebra A O ( ¯ X ω ). Definition 7.6 (The measure ¯ m ω ) . Let (cid:0) ( X i , d i , m i , p i ) (cid:1) be a sequence of pointed metricmeasure spaces. Then we define the set-function ¯ m ω : A O ( ¯ X ω ) → [0 , + ∞ ] as ¯ m ω (cid:0) (Π i → ω A i ) ∩ O ( ¯ X ω ) (cid:1) := lim R →∞ ˜ m ω (cid:0) Π i → ω A i ∩ B ( p i , R ) (cid:1) = lim R →∞ lim i → ω m i (cid:0) A i ∩ B ( p i , R ) (cid:1) for every Π i → ω A i ∈ A ( ¯ X ω ) . The set-function ¯ m ω is well-defined due to Lemma 7.2 and (7.3) . In order to show that ¯ m ω is consistent with ˜ m ω , we will need the following simple result. Lemma 7.7.
Let (cid:0) ( X i , d i , p i ) (cid:1) be a sequence of pointed metric spaces. Let ¯ A = Π i → ω A i be agiven element of A ( ¯ X ω ) ∩ A O ( ¯ X ω ) . Then the set ¯ A ⊂ O ( ¯ X ω ) is bounded with respect to ¯ d ω .Proof. We argue by contradiction: suppose that for every n ∈ N there exists [ x ni ] ∈ ¯ A suchthat lim i → ω d i ( x ni , p i ) = ¯ d ω (cid:0) [ x ni ] , [ p i ] (cid:1) > n . Define S := (cid:8) j ∈ S ( x i ) : d j ( x j , p j ) > (cid:9) and S n := (cid:8) j ∈ S n − ∩ S ( x ni ) (cid:12)(cid:12) d j ( x nj , p j ) > n (cid:9) , for every n ≥ . LTRALIMITS OF POINTED METRIC MEASURE SPACES 23
Observe that ( S n ) n ∈ N ⊂ ω by construction. Also, it holds that S n ⊂ S n − for every n ≥
2. Letus now set y i := x ni for every n ∈ N and i ∈ S n \ S n +1 . Notice that we have ( y i ) i ∈ S ∈ Q i ∈ S A i .Fix any n ∈ N . Given any i ∈ S n , there exists a unique n i ≥ n such that i ∈ S n i \ S n i +1 , thusaccordingly d i ( y i , p i ) = d i ( x n i i , p i ) > n i ≥ n . This shows that (cid:8) i ∈ S : d i ( y i , p i ) > n (cid:9) ∈ ω for all n ∈ N , in other words ¯ d ω (cid:0) [ y i ] , [ p i ] (cid:1) = lim i → ω d i ( y i , p i ) = + ∞ . This is in contradictionwith the fact that [ y i ] ∈ ¯ A ⊂ O ( ¯ X ω ), thus the statement is achieved. (cid:3) Thanks to the above lemma, we can now prove that ¯ m ω and ˜ m ω agree on A ( ¯ X ω ) ∩ A O ( ¯ X ω ). Proposition 7.8.
Let (cid:0) ( X i , d i , m i , p i ) (cid:1) be a sequence of pointed metric measure spaces. Thenthe set-function ¯ m ω is a finitely-additive measure on A O ( ¯ X ω ) . Moreover, it holds that ¯ m ω ( ¯ A ) = ˜ m ω ( ¯ A ) , for every ¯ A ∈ A ( ¯ X ω ) ∩ A O ( ¯ X ω ) . (7.4) Proof.
Let ¯ A, ¯ B ∈ A ( ¯ X ω ) be such that (cid:0) ¯ A ∩ O ( ¯ X ω ) (cid:1) ∩ (cid:0) ¯ B ∩ O ( ¯ X ω ) (cid:1) = ∅ , say that ¯ A = Π i → ω A i and ¯ B = Π i → ω B i . In particular, (cid:0) Π i → ω A i ∩ B ( p i , R ) (cid:1) ∩ (cid:0) Π i → ω B i ∩ B ( p i , R ) (cid:1) = ∅ for all R > m ω on A ( ¯ X ω ), we get that¯ m ω (cid:0)(cid:0) ¯ A ∩ O ( ¯ X ω ) (cid:1) ∪ (cid:0) ¯ B ∩ O ( ¯ X ω ) (cid:1)(cid:1) = ¯ m ω (cid:0) (Π i → ω A i ∪ B i ) ∩ O ( ¯ X ω ) (cid:1) = lim R →∞ ˜ m ω (cid:0) Π i → ω ( A i ∪ B i ) ∩ B ( p i , R ) (cid:1) = lim R →∞ ˜ m ω (cid:16)(cid:0) Π i → ω A i ∩ B ( p i , R ) (cid:1) ∪ (cid:0) Π i → ω B i ∩ B ( p i , R ) (cid:1)(cid:17) = lim R →∞ h ˜ m ω (cid:0) Π i → ω A i ∩ B ( p i , R ) (cid:1) + ˜ m ω (cid:0) Π i → ω B i ∩ B ( p i , R ) (cid:1)i = lim R →∞ ˜ m ω (cid:0) Π i → ω A i ∩ B ( p i , R ) (cid:1) + lim R →∞ ˜ m ω (cid:0) Π i → ω B i ∩ B ( p i , R ) (cid:1) = ¯ m ω (cid:0) ¯ A ∩ O ( ¯ X ω ) (cid:1) + ¯ m ω (cid:0) ¯ B ∩ O ( ¯ X ω ) (cid:1) , whence it follows that ¯ m ω is a finitely-additive measure. To prove (7.4), let ¯ A = Π i → ω A i be agiven element of A ( ¯ X ω ) ∩ A O ( ¯ X ω ). Lemma 7.7 grants that ¯ A is ¯ d ω -bounded, thus (recalling(7.2)) we can find R > i → ω A i ⊂ Π i → ω B ( p i , R ). By using Lemma 7.2 andProposition 7.3, we deduce that there exists S ∈ ω such that A i ⊂ B ( p i , R ) for every i ∈ S .Therefore, for any given R ′ > R we have that m i ( A i ) = m i (cid:0) A i ∩ B ( p i , R ′ ) (cid:1) for all i ∈ S , whichyields ˜ m ω (cid:0) Π i → ω A i ∩ B ( p i , R ′ ) (cid:1) = ˜ m ω ( ¯ A ). Then we can conclude that¯ m ω ( ¯ A ) = lim R ′ →∞ ˜ m ω (cid:0) Π i → ω A i ∩ B ( p i , R ′ ) (cid:1) = lim R ′ →∞ ˜ m ω ( ¯ A ) = ˜ m ω ( ¯ A ) , thus obtaining (7.4) and accordingly the statement. (cid:3) Hereafter, we shall work with sequences of pointed metric measure spaces satisfying thefollowing growth condition.
Definition 7.9 ( ω -uniform bounded finiteness) . Let (cid:0) ( X i , d i , m i , p i ) (cid:1) be a sequence of pointedmetric measure spaces. Then we say that (cid:0) ( X i , d i , m i , p i ) (cid:1) is a ω -uniformly boundedly finite sequence provided there exists η : (0 , + ∞ ) → (0 , + ∞ ) such that for any R > it holds that m i (cid:0) B ( p i , R ) (cid:1) ≤ η ( R ) , for ω -a.e. i. In the framework of ω -uniformly boundedly finite sequences of spaces, we will extend themeasure ¯ m ω to a σ -algebra ¯ B ω on O ( ¯ X ω ) containing the algebra A O ( ¯ X ω ). The σ -algebra ¯ B ω that we will consider is the one obtained by adding null sets to the elements of A O ( ¯ X ω ). Definition 7.10 (Null sets and ¯ B ω ) . Let (cid:0) ( X i , d i , m i , p i ) (cid:1) be a sequence of pointed metricmeasure spaces. Let N ⊂ O ( ¯ X ω ) be given. Then we declare that N is a null set provided forany ε > there exists ¯ A ∈ A O ( ¯ X ω ) such that N ⊂ ¯ A and ¯ m ω ( ¯ A ) < ε . The family of all nullsets is denoted by N ω . Moreover, we denote by ¯ B ω the family of all subsets A ⊂ O ( ¯ X ω ) forwhich the following property holds: given any R > , there exists ¯ A R ∈ A O ( ¯ X ω ) such that (cid:0) A ∩ Π i → ω B ( p i , R ) (cid:1) ∆ ¯ A R ∈ N ω . Definition 7.11 (The extension of ¯ m ω ) . Let (cid:0) ( X i , d i , m i , p i ) (cid:1) be a sequence of pointed metricmeasure spaces. Then we define ¯ m ω : ¯ B ω → [0 , + ∞ ] as follows: given any A ∈ ¯ B ω , we set ¯ m ω ( A ) := lim R →∞ ¯ m ω ( ¯ A R ) , (7.5) where the sets ¯ A R ∈ A O ( ¯ X ω ) are chosen so that (cid:0) A ∩ Π i → ω B ( p i , R ) (cid:1) ∆ ¯ A R ∈ N ω . It can be readily checked that ¯ m ω is well-defined ( i.e. , the limit in (7.5) does not dependon the choice of the sets ¯ A R ) and that it is consistent with Definition 7.6. Next we will show,under the ω -uniform bounded finiteness assumption, that ¯ B ω is a σ -algebra and that ¯ m ω is ameasure on ¯ B ω . We will need the following lemma, see [15, Lemma 2.3]. Lemma 7.12.
Let (cid:0) ( X i , d i , m i , p i ) (cid:1) be a ω -uniformly boundedly finite sequence of pointedmetric measure spaces. Let ( ¯ A n ) n ∈ N ⊂ A O ( ¯ X ω ) be given. Then it holds that S n ∈ N ¯ A n ∈ ¯ B ω .Proof. Let
R > n ∈ N we may write ¯ A n = (Π i → ω A ni ) ∩ O ( ¯ X ω ),where A ni ∈ Ball( X i ) for all i ∈ S n (for some S n ∈ ω ). Notice that [ n ∈ N ¯ A n ∩ (cid:0) Π i → ω B ( p i , R ) (cid:1) = [ n ∈ N (Π i → ω B ni ) ∩ B ( p i , R ) , where we set B ni := [ k ≤ n A ki . Therefore, we may assume that ¯ A n = Π i → ω A ni are internal measurable subsets of O ( ¯ X ω ) forwhich A ni ⊂ A mi ⊂ B ( p i , R ) whenever n ≤ m . Since m i (cid:0) B ( p i , R ) (cid:1) ≤ η ( R ) for ω -a.e. i , we maythus also assume that ¯ m ω ( ¯ A n ) = lim i → ω m i ( A ni ) ≤ η ( R ) < + ∞ .The rest of the proof is essentially the same as the proof of [15, Lemma 2.3]. We will presentit here for completeness. The procedure is a sort of diagonal argument. Denote M n := ¯ m ω ( ¯ A n )for all n ∈ N . Define the neighbourhood U n of M n as U n := [ M n − − n , M n + 2 − n ]. Definethe sets ( T n ) n ∈ N recursively as follows: T := (cid:8) i ∈ S : m i ( A i ) ∈ U (cid:9) and T n := (cid:8) i ∈ T n − ∩ S n (cid:12)(cid:12) i ≥ n, m i ( A ni ) ∈ U n (cid:9) , for every n ≥ . By definition of M n (and by induction), we have that T n ∈ ω for every n ∈ N . The sets T n are also nested by definition, i.e. , T n ⊂ T m whenever m ≤ n . For every index i ∈ T , there LTRALIMITS OF POINTED METRIC MEASURE SPACES 25 exists n i ∈ N so that i ∈ T n i \ T n i +1 . Define now C i := A n i i for every i ∈ T . Moreover, letus set ¯ C := Π i → ω C i ∈ A ( ¯ X ω ) ∩ A O ( ¯ X ω ). We claim that [ n ∈ N ¯ A n ⊂ ¯ C and ¯ C \ [ n ∈ N ¯ A n ∈ N ω , (7.6)whence the statement would immediately follow, thanks to the arbitrariness of R >
0. Toprove the first part of (7.6), let n ∈ N be fixed. Notice that if i ∈ T n , then n i ≥ n ; in turn, thisimplies that A ni ⊂ C i . In other words, T n = { i ∈ T n : A ni ⊂ C i } ∈ ω . This guarantees that¯ A n = Π i → ω A ni ⊂ Π i → ω C i = ¯ C , thus accordingly S n ∈ N ¯ A n ⊂ ¯ C . To prove the second part of(7.6), observe that ¯ A n = Π i → ω A ni ⊂ Π i → ω A mi = ¯ A m whenever n ≤ m , thus ( M n ) n ⊂ (cid:2) , η ( R ) (cid:3) is a non-decreasing sequence, which admits a limit M ∞ := lim n →∞ M n ∈ (cid:2) , η ( R ) (cid:3) . Let ε > N ∈ N such that 2 − N < ε and | M n − M ∞ | < ε for every n ≥ N . Since T N = (cid:8) i ∈ T N (cid:12)(cid:12) n i ≥ N (cid:9) = (cid:8) i ∈ T N (cid:12)(cid:12) | m i ( C i ) − M n i | < ε (cid:9) ∩ (cid:8) i ∈ T N (cid:12)(cid:12) | M n i − M ∞ | < ε (cid:9) = (cid:8) i ∈ T N (cid:12)(cid:12) | m i ( C i ) − M ∞ | < ε (cid:9) ∈ ω, we get that ¯ m ω ( ¯ C ) = lim i → ω m i ( C i ) = M ∞ . Hence, we have that¯ m ω (cid:0) ¯ C \ S k ≤ n ¯ A k (cid:1) = ¯ m ω ( ¯ C \ ¯ A n ) = ¯ m ω ( ¯ C ) − ¯ m ω ( ¯ A n ) = M ∞ − M n −→ , as n → ∞ . Given that ¯ C \ S k ∈ N ¯ A k ⊂ ¯ C \ S k ≤ n ¯ A k for all n ∈ N , we conclude that ¯ C \ S n ∈ N ¯ A n ∈ N ω . (cid:3) Theorem 7.13.
Let (cid:0) ( X i , d i , m i , p i ) (cid:1) be a ω -uniformly boundedly finite sequence of pointedmetric measure spaces. Then the family ¯ B ω is a σ -algebra and the set-function ¯ m ω is a(countably-additive) measure on ¯ B ω .Proof. Since ∅ ∈ N ω ∩ A O ( ¯ X ω ), we have that ∅ ∈ ¯ B ω . To check that ¯ B ω is a σ -algebra, noticethat if A, B ∈ ¯ B ω , then for any given R > A R , ¯ B R ∈ A O ( ¯ X ω ) such that A ′ ∆ ¯ A R , B ′ ∆ ¯ B R ∈ N ω , where A ′ := A ∩ Π i → ω B ( p i , R ) and B ′ := B ∩ Π i → ω B ( p i , R ) . Therefore, we deduce that ( ¯ A R \ ¯ B R )∆( A ′ \ B ′ ) ⊂ ( ¯ A R ∆ A ′ )∆( ¯ B R ∆ B ′ ) ∈ N ω , thus provingthat A \ B ∈ ¯ B ω by arbitrariness of R >
0. Let us now show that A := S n ∈ N A n ∈ ¯ B ω whenever ( A n ) n ∈ N ⊂ ¯ B ω . Let R > n ∈ N , pick some set ¯ A nR ∈ A O ( ¯ X ω )such that ( C R ∩ A n )∆ ¯ A nR ∈ N ω , where we put C R := Π i → ω B ( p i , R ). Then by Lemma 7.12there exists ¯ A R ∈ A O ( ¯ X ω ) such that (cid:0) C R ∩ S n ∈ N ¯ A nR (cid:1) ∆ ¯ A R ∈ N ω . Now observe that( C R ∩ A )∆ ¯ A R ⊂ (cid:0) ( C R ∩ A )∆( C R ∩ S n ∈ N ¯ A nR ) (cid:1) ∪ (cid:0) ( C R ∩ S n ∈ N ¯ A nR )∆ ¯ A R (cid:1) ⊂ (cid:0)S n ∈ N ( C R ∩ A n )∆ ¯ A nR (cid:1) ∪ (cid:0) ( C R ∩ S n ∈ N ¯ A nR )∆ ¯ A R (cid:1) . Using Lemma 7.12 it is easy to check that countable unions of null sets are null sets, thus( C R ∩ A )∆ ¯ A R ∈ N ω and so A ∈ ¯ B ω . All in all, we have proven that ¯ B ω is a σ -algebra.Let us now show that ¯ m ω is a measure on ¯ B ω . Suppose that ( A n ) n ∈ N ⊂ ¯ B ω are pairwisedisjoint sets. Define A := S n ∈ N A n . We aim to prove that¯ m ω ( A ) = X n ∈ N ¯ m ω ( A n ) . (7.7) First of all, fix any
R > A R , ¯ A nR ∈ A O ( ¯ X ω ) such that (cid:0) A ∩ Π i → ω B ( p i , R ) (cid:1) ∆ ¯ A R and (cid:0) A n ∩ Π i → ω B ( p i , R ) (cid:1) ∆ ¯ A nR belong to N ω . Observe that¯ m ω (cid:0) ¯ A R \ S n ≤ N ¯ A nR (cid:1) = ¯ m ω ( ¯ A R ) − N X n =1 ¯ m ω ( ¯ A nR ) −→ , as N → ∞ . (7.8)By using the fact that ¯ m ω ( ¯ A R ) ≥ P Nn =1 ¯ m ω ( ¯ A nR ), we can deduce that¯ m ω ( A ) = lim R →∞ ¯ m ω ( ¯ A R ) ≥ lim R →∞ N X n =1 ¯ m ω ( ¯ A nR ) = N X n =1 ¯ m ω ( A n ) −→ X n ∈ N ¯ m ω ( A n ) , as N → ∞ , thus ¯ m ω ( A ) ≥ P n ∈ N ¯ m ω ( A n ). Let us prove the other inequality. If P n ∈ N ¯ m ω ( A n ) = + ∞ ,then the inequality trivially holds. Suppose now that P n ∈ N ¯ m ω ( A n ) < + ∞ . Given any k ∈ N ,it follows from (7.8) that there exists N k ∈ N such that ¯ m ω ( ¯ A k ) − P n ≤ N k ¯ m ω ( ¯ A nk ) ≤ − k .Then ¯ m ω ( ¯ A k ) − k ≤ X n ≤ N k ¯ m ω ( ¯ A nk ) ≤ X n ∈ N lim k ′ →∞ ¯ m ω ( ¯ A nk ′ ) = X n ∈ N ¯ m ω ( A n ) . By letting k → ∞ we conclude that (7.7) holds, whence the statement follows. (cid:3) So far we have defined a natural measure on O ( ¯ X ω ). Now we have everything that weneed to define the measure on the ultralimit of pointed metric spaces, and hence to define theultralimit of pointed metric measure spaces. Definition 7.14 (Ultralimit of pointed metric measure spaces) . Let (cid:0) ( X i , d i , m i , p i ) (cid:1) be a ω -uniformly boundedly finite sequence of pointed metric measure spaces. Let us consider the σ -algebra B ω := π ∗ ¯ B ω on X ω , where the projection π : O ( ¯ X ω ) → X ω is defined as in (6.2) .Given that Ball( X ω ) ⊂ B ω by Proposition 7.5, we can define the ball measure m ω on X ω as m ω := ( π ∗ ¯ m ω ) | Ball( X ω ) . (7.9) Then the ultralimit of the sequence (cid:0) ( X i , d i , m i , p i ) (cid:1) is defined as lim i → ω ( X i , d i , m i , p i ) := ( X ω , d ω , m ω , p ω ) . Notice that, by the following reasoning, the previous definition makes sense. Given any[ x i ] ∈ O ( ¯ X ω ) and R >
0, it clearly holds that Π i → ω B ( x i , R + 1 /n ) ∈ A O ( ¯ X ω ) for every n ∈ N ,thus we have ¯ B (cid:0) [ x i ] , R (cid:1) ∈ σ (cid:0) A O ( ¯ X ω ) (cid:1) ⊂ ¯ B ω by (7.2). This grants that Ball (cid:0) O ( ¯ X ω ) (cid:1) ⊂ ¯ B ω .Moreover, it is immediate to check that B (cid:0) [ x i ] , R (cid:1) = π − (cid:0) B (cid:0) [[ x i ]] , R (cid:1)(cid:1) for all [ x i ] ∈ O ( ¯ X ω )and R >
0. In particular, Ball( X ω ) ⊂ π ∗ Ball (cid:0) O ( ¯ X ω ) (cid:1) . Given that Ball (cid:0) O ( ¯ X ω ) (cid:1) ⊂ ¯ B ω , wefinally conclude that Ball( X ω ) ⊂ π ∗ ¯ B ω = B ω , thus the definition (7.9) makes sense. Remark 7.15.
Here we have chosen to define the ultralimit as a metric measure space insuch a way that the measure is considered as a ball measure. This choice is to make thingsmore consistent and clean, and is suitable for our purposes. However, in some cases, as inExample 7.16, this leads to a loss of information about the measure. We point out that onemight want to consider the measure m ω as a measure on some larger σ -algebra instead. (cid:4) LTRALIMITS OF POINTED METRIC MEASURE SPACES 27
The following construction and its variants will play a central role in the rest of the paper.The example shows, for instance, that the ball σ -algebra of the ultralimit space X ω might bemuch smaller than B ω , thus a fortiori also of the Borel σ -algebra of X ω . Example 7.16.
Given any i ∈ N , consider the pointed metric measure space ( X i , d i , m i , p i ),which is defined as follows: the set X i is made of i distinct points x i , . . . , x ii , the distance d i isgiven by d i ( x ij , x ij ′ ) := 1 for every j, j ′ = 1 , . . . , i with j = j ′ , the measure m i is the uniformlydistributed probability measure m i := i P ij =1 δ x ij , and p i := x i .Being (( X i , d i , m i , p i ) (cid:1) a ω -uniformly boundedly finite sequence (trivially, since each m i isa probability measure), its ultralimit ( X ω , d ω , m ω , p ω ) = lim i → ω ( X i , d i , m i , p i ) exists. One cancheck that X ω is an uncountable set and d ω ( x, y ) = 1 for every x, y ∈ X ω with x = y . Then B ( x, r ) = ( { x } X ω if r ≤ , if r > . (7.10)In particular, every singleton in X ω is an open ball, so that B ( X ω ) = 2 X ω (since every subsetof X ω is a union of singletons). On the other hand, the ball σ -algebra on X ω is given byBall( X ω ) = (cid:8) A ⊂ X ω (cid:12)(cid:12) A is either countable or cocountable (cid:9) , (7.11)where by cocountable we mean that its complement X ω \ A is countable. This identity can beeasily verified: since Ball( X ω ) contains all singletons and is a σ -algebra, it includes the familyin the right-hand side of (7.11). Being the latter a σ -algebra, we conclude that (7.11) holds.It can be readily checked that the limit measure m ω is given by m ω ( A ) = (
01 if A ∈ Ball( X ω ) is countable,if A ∈ Ball( X ω ) is cocountable. (7.12)In particular, m ω is a probability measure. (cid:4) Remark 7.17.
As pointed out in Remark 7.15, we lose information on the measure m ω whenwe restrict it to the ball σ -algebra. Indeed, while m ω achieves only values 0 and 1 as a ballmeasure, it achieves all the values between 0 and 1 when regarded as a measure on B ω . (cid:4) Lemma 7.18.
Let (cid:0) ( X i , d i , m i , p i ) (cid:1) be a ω -uniformly boundedly finite sequence of pointedmetric measure spaces. Then the ultralimit measure m ω is boundedly finite.Proof. It suffices to prove that m ω (cid:0) B ( p ω , R ) (cid:1) is finite for every R >
0, since every ball iscontained in B ( p ω , R ) for R >
R > R ′ > R , weknow from (7.2) that B (cid:0) [ p i ] , R (cid:1) ⊂ Π i → ω B ( p i , R ′ ). Therefore, we have that m ω (cid:0) B ( p ω , R ) (cid:1) = ¯ m ω (cid:0) B (cid:0) [ p i ] , R (cid:1)(cid:1) ≤ ¯ m ω (cid:0) Π i → ω B ( p i , R ′ ) (cid:1) = lim i →∞ m i (cid:0) B ( p i , R ′ ) (cid:1) ≤ η ( R ′ ) < + ∞ , whence the statement follows. (cid:3) Remark 7.19.
Observe that if B ( X ω ) and Ball( X ω ) coincide (which happens, for instance,when X ω is separable), then m ω is a Borel measure on X ω . On the other hand, in general itholds that Ball( X ω ) and B ω might fail to contain B ( X ω ). (cid:4) On the support-concentration issue
As mentioned in Remark 7.19, the ultralimit of metric measure spaces might fail to beseparable. In some cases, however, the measure is still concentrated on a separable set andhence it might be thought of as a Borel measure on its support. In this section, we willcharacterise those sequences for which the ultralimit measure is concentrated on a separableset. For that we will need the following definitions of measure-theoretic variants of totalboundedness and asymptotic total boundedness.
Definition 8.1 (Bounded m -total boundedness) . Let ( X, d, m , p ) be a pointed, boundedlyfinite metric measure space. Then we say that ( X, d, m , p ) is boundedly m -totally bounded provided for every R, r, ε > there exists a finite collection of points ( x n ) Mn =1 ⊂ X such that m (cid:0) ¯ B ( p, R ) \ S Mn =1 B ( x n , r ) (cid:1) ≤ ε. Definition 8.2 (Asymptotic bounded m ω -total boundedness) . Let (cid:0) ( X i , d i , m i , p i ) (cid:1) be a ω -uniformly boundedly finite sequence of pointed metric measure spaces. Then we say that thesequence (cid:0) ( X i , d i , m i , p i ) (cid:1) is asymptotically boundedly m ω -totally bounded provided for everygiven R, r, ε > there exist a number M ∈ N and points ( x in ) Mn =1 ⊂ X i such that lim i → ω m i (cid:0) ¯ B ( p i , R ) \ S Mn =1 B ( x in , r ) (cid:1) ≤ ε. With these definitions in hand, we may state the main result of this section.
Theorem 8.3.
Let (cid:0) ( X i , d i , m i , p i ) (cid:1) be a ω -uniformly boundedly finite sequence of pointedmetric measure spaces. Then the following conditions are equivalent: i) m ω is concentrated on spt( m ω ) . ii) m ω is concentrated on a closed and separable set C ⊂ X ω . (Observe that Remark 3.3grants that C ∈ Ball( X ω ) .) iii) The ultralimit ( X ω , d ω , m ω , p ω ) is boundedly m ω -totally bounded. iv) The sequence (cid:0) ( X i , d i , m i , p i ) (cid:1) is asymptotically boundedly m ω -totally bounded.Proof. i) = ⇒ ii) It follows from the fact that spt( m ω ) is closed and separable, as shown in Lemma3.2. (Observe that the σ -finiteness of the measure m ω is granted by Lemma 7.18.)ii) = ⇒ i) Suppose m ω is concentrated on a closed, separable set C ∈ Ball( X ω ). If C ⊂ spt( m ω ),then m ω is concentrated on spt( m ω ). Now suppose C \ spt( m ω ) = ∅ . Given any x ∈ C \ spt( m ω ),by definition of support we can find a radius r x > m ω (cid:0) B ( x, r x ) (cid:1) = 0. Being C separable, we can use Lindel¨of’s lemma to find a sequence ( x n ) n ∈ N ⊂ C \ spt( m ω ) such that C \ spt( m ω ) ⊂ S n ∈ N B ( x n , r x n ). Therefore, we can conclude that m ω (cid:0) X \ spt( m ω ) (cid:1) ≤ m ω ( X \ C ) + m ω (cid:0) C \ spt( m ω ) (cid:1) ≤ X n ∈ N m ω (cid:0) B ( x n , r x n ) (cid:1) = 0 , which shows that m ω is concentrated on spt( m ω ).ii) = ⇒ iii) Suppose m ω is concentrated on a closed, separable set C ∈ Ball( X ω ). Let R, r, ε > LTRALIMITS OF POINTED METRIC MEASURE SPACES 29 be fixed. Choose a dense sequence ( y n ) n ∈ N in C . Then it holds ¯ B ( p ω , R ) ∩ C ⊂ S n ∈ N B ( y n , r ).By using the continuity from above of m ω and the fact that ¯ B ( p ω , R ) has finite m ω -measure,we deduce thatlim M →∞ m ω (cid:0) ¯ B ( p ω , R ) \ S Mn =1 B ( y n , r ) (cid:1) = lim M →∞ m ω (cid:0)(cid:0) ¯ B ( p ω , R ) ∩ C (cid:1) \ S Mn =1 B ( y n , r ) (cid:1) = 0 . Therefore, there exists M ∈ N such that m ω (cid:0) ¯ B ( p ω , R ) \ S Mn =1 B ( y n , r ) (cid:1) ≤ ε , thus proving thatthe ultralimit ( X ω , d ω , m ω , p ω ) is boundedly m ω -totally bounded.iii) = ⇒ ii) Suppose ( X ω , d ω , m ω , p ω ) is boundedly m ω -totally bounded. Given any i, j, ℓ ∈ N ,choose any M ijℓ ∈ N and ( x ijℓn ) M ijℓ n =1 ⊂ X ω such that m ω (cid:16) ¯ B ( p ω , i ) \ S M ijℓ n =1 B (cid:0) x ijℓn , /j (cid:1)(cid:17) ≤ ℓ . Let us define the closed and separable set C ⊂ X ω as C := [ i,j,ℓ ∈ N (cid:8) x ijℓ , . . . , x ijℓM ijℓ (cid:9) . We claim that m ω is concentrated on C . We argue by contradiction: suppose there is i ∈ N such that m ω (cid:0) ¯ B ( p ω , i ) \ C (cid:1) >
0. Since C is closed and separable, every ε -neighbourhood C ε of C can be written as a countable union of balls, thus in particular C ε ∈ Ball( X ω ). Since C isthe intersection of its neighbourhoods, there is j ∈ N such that δ := m ω (cid:0) ¯ B ( p ω , i ) \ C /j (cid:1) > ℓ ∈ N such that 1 /ℓ ≤ δ/
2. Then it holds that δ = m ω (cid:0) ¯ B ( p ω , i ) \ C /j (cid:1) ≤ m ω (cid:16) ¯ B ( p ω , i ) \ S M i j ℓ n =1 B (cid:0) x i j ℓ n , /j (cid:1)(cid:17) ≤ ℓ ≤ δ < δ, which leads to a contradiction. Consequently, m ω is concentrated on C , as required.iii) = ⇒ iv) Suppose ( X ω , d ω , m ω , p ω ) is boundedly m ω -totally bounded. Let R, r, ε > R ′ > R and r ′ ∈ (0 , r ). Choose finitely many points (cid:0) [[ x ni ]] (cid:1) Mn =1 ⊂ X ω such that m ω (cid:16) ¯ B ( p ω , R ′ ) \ S Mn =1 B (cid:0) [[ x ni ]] , r ′ (cid:1)(cid:17) ≤ ε. Then it holds thatlim i → ω m i (cid:0) ¯ B ( p i , R ) \ S Mn =1 B ( x ni , r ) (cid:1) ≤ m ω (cid:16) ¯ B ( p ω , R ′ ) \ S Mn =1 B (cid:0) [[ x ni ]] , r ′ (cid:1)(cid:17) ≤ ε, thus showing that (cid:0) ( X i , d i , m i , p i ) (cid:1) is asymptotically boundedly m ω -totally bounded.iv) = ⇒ iii) Suppose the sequence (cid:0) ( X i , d i , m i , p i ) (cid:1) is asymptotically boundedly m ω -totallybounded. Let R, r, ε > R ′ > R and r ′ ∈ (0 , r ). Then there exist anumber M ∈ N and points ( x ni ) Mn =1 ⊂ X i such thatlim i → ω m i (cid:0) ¯ B ( p i , R ′ ) \ S Mn =1 B ( x ni , r ′ ) (cid:1) ≤ ε. Define x n := [[ x ni ]] ∈ X ω for every n = 1 , . . . , M . Therefore, it holds that m ω (cid:0) ¯ B ( p ω , R ) \ S Mn =1 B ( x n , r ) (cid:1) = inf k ∈ N lim i → ω m i (cid:16) ¯ B (cid:0) p i , R + 1 /k (cid:1) \ S Mn =1 B (cid:0) x ni , r − /k (cid:1)(cid:17) ≤ lim i → ω m i (cid:0) ¯ B ( p i , R ′ ) \ S Mn =1 B ( x ni , r ′ ) (cid:1) ≤ ε, thus proving that ( X ω , d ω , m ω , p ω ) is boundedly m ω -totally bounded. (cid:3) Example 8.4.
Let (cid:0) ( X i , d i , m i , p i ) (cid:1) be as in Example 7.16. Let us show that (cid:0) ( X i , d i , m i , p i ) (cid:1) is not asymptotically boundedly m ω -totally bounded: observe that for any i ∈ N it holds that F ⊂ X i , m i (cid:0) X i \ S x ∈ F B ( x, (cid:1) ≤
12 = ⇒ F ≥ i , which proves that (cid:0) ( X i , d i , m i , p i ) (cid:1) fails to be asymptotically boundedly m ω -totally boundedfor the choice ( R, r, ε ) = (2 , , / m ω (cid:0) B ( x, (cid:1) = m ω (cid:0) { x } (cid:1) = 0 for every x ∈ X ω by (7.10) and (7.12), it holdsthat spt( m ω ) = ∅ and thus m ω is not concentrated on spt( m ω ). (cid:4) Lemma 8.5.
Let (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ ¯ N be a sequence of pointed Polish metric measure spaces,with m i boundedly finite. Suppose ( X i , d i , m i , p i ) → ( X ∞ , d ∞ , m ∞ , p ∞ ) in the pmGH sense.Then it holds that the sequence (cid:0) ( X i , d i , m i , p i ) (cid:1) is ω -uniformly boundedly finite.Proof. Fix ( R i , ε i )-approximations ψ i : X i → X ∞ as in Definition 4.6. We claim thatlim i →∞ m i (cid:0) B ( p i , R ) (cid:1) ≤ m ∞ (cid:0) ¯ B ( p ∞ , R ) (cid:1) , for every R > . (8.1)Let R > f k ) k ⊂ C b (cid:0) B ( p ∞ , R + 1) (cid:1) with 0 ≤ f k ≤ (cid:0) { f k < } , ¯ B ( p ∞ , R ) (cid:1) > f k → χ ¯ B ( p ∞ ,R ) . Fix k ∈ N , then choose any j ∈ N such that R i > R , ε i <
1, and f k = 1 on B ( p ∞ , R + ε i ) for all i ≥ j . Since ψ i is a ( R i , ε i )-approximation,we see that ψ i (cid:0) B ( p i , R ) (cid:1) ⊂ B ( p ∞ , R + ε i ), thus B ( p i , R ) ⊂ ψ − i (cid:0) B ( p ∞ , R + ε i ) (cid:1) and ˆ f k d m ∞ = lim i →∞ ˆ f k d( ψ i ) ∗ m i = lim i →∞ ˆ f k ◦ ψ i d m i ≥ lim i →∞ ˆ χ B ( p ∞ ,R + ε i ) ◦ ψ i d m i = lim i →∞ m i (cid:0) ψ − i (cid:0) B ( p ∞ , R + ε i ) (cid:1)(cid:1) ≥ lim i →∞ m i (cid:0) B ( p i , R ) (cid:1) . By letting k → ∞ and using the dominated convergence theorem, we get (8.1). Now let usdefine the function η : (0 , + ∞ ) → [1 , + ∞ ) as η ( R ) := m ∞ (cid:0) ¯ B ( p ∞ , R ) (cid:1) + 1 for every R > m ∞ is boundedly finite). Then (8.1) implies that forany R > m i (cid:0) B ( p i , R ) (cid:1) ≤ η ( R ) for ω -a.e. i , which shows that (cid:0) ( X i , d i , m i , p i ) (cid:1) is ω -uniformly boundedly finite. (cid:3) Proposition 8.6.
Let (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ ¯ N be a sequence of pointed Polish metric measurespaces, with m i boundedly finite. Suppose ( X i , d i , m i , p i ) → ( X ∞ , d ∞ , m ∞ , p ∞ ) in the pmGHsense. Then the sequence (cid:0) ( X i , d i , m i , p i ) (cid:1) is asymptotically boundedly m ω -totally bounded.In particular, it holds that m ω is concentrated on spt( m ω ) .Proof. Fix ( R i , ε i )-approximations ψ i : X i → X ∞ as in Definition 4.6. Let φ i : X ∞ → X i bea quasi-inverse of ψ i . Fix any R, r, ε >
0. Being m ∞ boundedly finite and X ∞ separable, wecan find M ∈ N and points ( x n ∞ ) Mn =1 ⊂ ¯ B ( p ∞ , R ) such that m ∞ ( C ∞ ) ≤ ε, where we set C ∞ := ¯ B ( p ∞ , R ) \ M [ n =1 B ( x n ∞ , r ) . LTRALIMITS OF POINTED METRIC MEASURE SPACES 31
Fix an open, bounded set Ω ⊂ X ∞ with C ∞ ⊂ Ω and a sequence ( f k ) k ⊂ C b (Ω) with0 ≤ f k ≤ (cid:0) { f k < } , C ∞ (cid:1) > f k → χ C ∞ . We define the sets C i ⊂ X i as C i := ¯ B ( p i , R ) \ M [ n =1 B (cid:0) φ i ( x n ∞ ) , r (cid:1) , for every i ∈ N . Fix k ∈ N , then choose any j ∈ N such that f k = 1 on ¯ B ( p ∞ , R + ε i ) \ S Mn =1 B ( x n ∞ , r − ε i ), r + R < R i + ε i , and r > ε i for all i ≥ j . Being the map ψ i a ( R i , ε i )-approximation,we see that ¯ B ( p i , R ) ⊂ ψ − i (cid:0) ¯ B ( p ∞ , R + ε i ) (cid:1) . Moreover, by using (4.3b) we can deduce that ψ − i (cid:0) B ( x n ∞ , r − ε i ) (cid:1) ⊂ B (cid:0) φ i ( x n ∞ ) , r (cid:1) . All in all, we have proven that for any i ≥ j it holds C i ⊂ ψ − i ( ˜ C i ) , where we set ˜ C i := ¯ B ( p ∞ , R + ε i ) \ M [ n =1 B ( x n ∞ , r − ε i ) . Therefore, we conclude that ˆ f k d m ∞ = lim i →∞ ˆ f k d( ψ i ) ∗ m i = lim i →∞ ˆ f k ◦ ψ i d m i ≥ lim i →∞ ˆ χ ˜ C i ◦ ψ i d m i = lim i →∞ m i (cid:0) ψ − i ( ˜ C i ) (cid:1) ≥ lim i →∞ m i ( C i ) . By letting k → ∞ and using the dominated convergence theorem, we thus obtain thatlim i → ω m i ( C i ) ≤ lim i →∞ m i ( C i ) ≤ m ∞ ( C ∞ ) ≤ ε, showing that (cid:0) ( X i , d i , m i , p i ) (cid:1) is an asymptotically boundedly m ω -totally bounded sequence,as required. The last part of the statement now follows from Theorem 8.3. (cid:3) Relation with the pmGH convergence
Aim of this section is to investigate the relation between the pointed (measured) Gromov–Hausdorff convergence and the ultralimits of pointed metric (measure) spaces.First of all, let us prove the following technical result, which will be needed in the sequel.
Lemma 9.1.
Let ( X, d ) be a complete metric space. Let ( x i ) i ∈ N ⊂ X be a sequence for whichthe ultralimit lim i → ω x i does not exist. Then there exists ε > with the following property:given any S ∈ ω , the set { x i } i ∈ S cannot be covered by finitely many balls of radius ε .Proof. We argue by contradiction: suppose there exist ε k ց S k ) k ∈ N ⊂ ω such that { x i } i ∈ S k ⊂ n k [ n =1 B ( x kn , ε k ) , for some n k ∈ N and { x kn } n k n =1 ⊂ { x i } i ∈ S k . In particular, by using the upward-closedness of ω , we see that for every k ∈ N it holds n k [ n =1 (cid:8) i ∈ N (cid:12)(cid:12) x i ∈ B ( x kn , ε k ) (cid:9) = (cid:26) i ∈ N (cid:12)(cid:12)(cid:12)(cid:12) x i ∈ n k [ n =1 B ( x kn , ε k ) (cid:27) ∈ ω. Therefore, there must exist ˜ x k ∈ { x kn } n k n =1 such that (cid:8) i ∈ N (cid:12)(cid:12) x i ∈ B (˜ x k , ε k ) (cid:9) ∈ ω . Now let usrecursively define ( T k ) k ∈ N ⊂ ω as follows: T := (cid:8) i ∈ N (cid:12)(cid:12) d ( x i , ˜ x ) < ε (cid:9) and T k +1 := (cid:8) i ∈ T k (cid:12)(cid:12) d ( x i , ˜ x k ) < ε k (cid:9) , for every k ∈ N . We claim that the sequence (˜ x k ) k ∈ N ⊂ X is Cauchy. Observe that for any k, k , k ∈ N suchthat k ≤ k ≤ k and j ∈ T k , we have that d (˜ x k , ˜ x k ) ≤ d (˜ x k , x j ) + d ( x j , ˜ x k ) ≤ ε k + ε k ≤ ε k . This implies that lim k →∞ sup (cid:8) d (˜ x k , ˜ x k ) : k , k ≥ k (cid:9) ≤ k →∞ ε k = 0, getting the claim.Given that ( X, d ) is complete, the limit x := lim k →∞ ˜ x k ∈ X exists. Now fix any ε > k ∈ N such that ε k < ε/ d ( x, ˜ x k ) < ε/
2. Hence, we have that (cid:8) i ∈ N (cid:12)(cid:12) d ( x i , x ) < ε (cid:9) ⊃ (cid:8) i ∈ N (cid:12)(cid:12) d ( x i , ˜ x k ) < ε/ (cid:9) ⊃ (cid:8) i ∈ N (cid:12)(cid:12) d ( x i , ˜ x k ) < ε k (cid:9) ∈ ω. This grants that (cid:8) i ∈ N (cid:12)(cid:12) d ( x i , x ) < ε (cid:9) ∈ ω for every ε >
0, which shows that the ultralimitlim i → ω x i ∈ X exists (and coincides with x ), thus leading to a contradiction. (cid:3) Theorem 9.2.
Let (cid:8) ( X i , d i , p i ) (cid:9) i ∈ ¯ N be a sequence of pointed Polish metric spaces. Suppose ( X i , d i , p i ) → ( X ∞ , d ∞ , p ∞ ) in the pGH sense. Let ψ i : X i → X ∞ be ( R i , ε i ) -approximations,where R i ր + ∞ and ε i ց , with quasi-inverses φ i : X ∞ → X i . Define φ ∞ : X ∞ → X ω as φ ∞ ( x ) := (cid:2)(cid:2) φ i ( x ) (cid:3)(cid:3) , for every x ∈ X ∞ . Then φ ∞ is an isometric embedding such that φ ∞ ( p ∞ ) = p ω . The map ψ ∞ : φ ∞ ( X ∞ ) → X ∞ ,which is given by ψ ∞ (cid:0) [[ x i ]] (cid:1) := lim i → ω ψ i ( x i ) , for every [[ x i ]] ∈ φ ∞ ( X ∞ ) , is well-defined and is the inverse of φ ∞ : X ∞ → φ ∞ ( X ∞ ) . Moreover, it holds that ( φ ∞ ) ∗ B ( X ∞ ) = B (cid:0) φ ∞ ( X ∞ ) (cid:1) ⊂ Ball( X ω ) . (9.1) Proof.
Let x, y ∈ X ∞ be fixed. Pick any j ∈ N such that x, y ∈ B ( p ∞ , R i − ε i ) and 4 ε i < R i for every i ≥ j . Then it holds (cid:12)(cid:12)(cid:12) d ∞ ( x, y ) − d i (cid:0) φ i ( x ) , φ i ( y ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ d ∞ (cid:0) x, ( ψ i ◦ φ i )( x ) (cid:1) + d ∞ (cid:0) y, ( ψ i ◦ φ i )( y ) (cid:1) + (cid:12)(cid:12)(cid:12) d ∞ (cid:0) φ i ( x ) , φ i ( y ) (cid:1) − d ∞ (cid:0) ( ψ i ◦ φ i )( x ) , ( ψ i ◦ φ i )( y ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ ε i , for every i ≥ j , whence it follows that d ∞ ( x, y ) = lim i → ω d i (cid:0) φ i ( x ) , φ i ( y ) (cid:1) = d ω (cid:0) φ ∞ ( x ) , φ ∞ ( y ) (cid:1) . This shows that φ ∞ is an isometry. Now let [[ x i ]] ∈ φ ∞ ( X ∞ ) be fixed. Pick any x ∈ X ∞ forwhich φ ∞ ( x ) = [[ x i ]]. Since [ x i ] ∈ O ( ¯ X ω ), one has lim i → ω d i ( x i , p i ) = ¯ d ω (cid:0) [ x i ] , [ p i ] (cid:1) < + ∞ ,thus there exist R > S ∈ ω such that d i ( x i , p i ) ≤ R for every i ∈ S . Fix any j ∈ N such that R < R i − ε i and x ∈ B ( p ∞ , R i − ε i ) for every i ≥ j . Notice that d ∞ (cid:0) ψ i ( x i ) , p ∞ (cid:1) ≤ d i ( x i , p i ) + ε i ≤ R + ε i < R i − ε i , for every i ∈ S with i ≥ j, LTRALIMITS OF POINTED METRIC MEASURE SPACES 33 so that ψ i ( x i ) ∈ B ( p ∞ , R i − ε i ). Therefore, for ω -a.e. i it holds that d ∞ (cid:0) ψ i ( x i ) , x (cid:1) ≤ d i (cid:0) ( φ i ◦ ψ i )( x i ) , φ i ( x ) (cid:1) + 3 ε i ≤ d i (cid:0) φ i ( x ) , x i (cid:1) + d i (cid:0) x i , ( φ i ◦ ψ i )( x i ) (cid:1) + 3 ε i ≤ d i (cid:0) φ i ( x ) , x i (cid:1) + 6 ε i , thus accordingly lim i → ω d ∞ (cid:0) ψ i ( x i ) , x (cid:1) ≤ lim i → ω d i (cid:0) φ i ( x ) , x i (cid:1) = ¯ d ω (cid:0)(cid:2) φ i ( x ) (cid:3) , [ x i ] (cid:1) = 0. Thisshows that the ultralimit ψ ∞ (cid:0) [[ x i ]] (cid:1) := lim i → ω ψ i ( x i ) exists and coincides with x , in otherwords ( ψ ∞ ◦ φ ∞ )( x ) = x . Then ψ ∞ : φ ∞ ( X ∞ ) → X ∞ is the inverse of φ ∞ : X ∞ → φ ∞ ( X ∞ ).Finally, let us prove (9.1). Given that φ ∞ is an isometric bijection between the metricspaces ( X ∞ , d ∞ ) and (cid:0) φ ∞ ( X ∞ ) , d ω | φ ∞ ( X ∞ ) × φ ∞ ( X ∞ ) (cid:1) , it holds ( φ ∞ ) ∗ B ( X ∞ ) = B (cid:0) φ ∞ ( X ∞ ) (cid:1) .Moreover, φ ∞ ( X ∞ ) is a closed separable subset of X ω , thus every closed subset C of φ ∞ ( X ∞ )(which is a fortiori closed in X ω ) belongs to Ball( X ω ) by Remark 3.3. Since B (cid:0) φ ∞ ( X ∞ ) (cid:1) isgenerated by the closed subsets of φ ∞ ( X ∞ ), we can conclude that (9.1) is verified. (cid:3) Corollary 9.3.
Let (cid:8) ( X i , d i , p i ) (cid:9) i ∈ ¯ N be a sequence of pointed Polish metric spaces. Suppose ( X i , d i , p i ) → ( X ∞ , d ∞ , p ∞ ) in the pGH sense. Let ψ i : X i → X ∞ be ( R i , ε i ) -approximations,with quasi-inverses φ i : X ∞ → X i . Let φ ∞ : X ∞ → X ω be as in Theorem 9.2. Then anelement [[ x i ]] ∈ X ω belongs to φ ∞ ( X ∞ ) if and only if the ultralimit x := lim i → ω ψ i ( x i ) ∈ X ∞ exists. In this case, it holds that φ ∞ ( x ) = [[ x i ]] . In particular, if ( X ∞ , d ∞ ) is a proper metricspace, then φ ∞ ( X ∞ ) = X ω and accordingly φ ∞ is an isomorphism of pointed metric spaces.Proof. Let [[ x i ]] ∈ X ω be given. If [[ x i ]] ∈ φ ∞ ( X ∞ ), then we know from Theorem 9.2 thatthe ultralimit ψ ∞ (cid:0) [[ x i ]] (cid:1) = lim i → ω ψ i ( x i ) ∈ X ∞ exists and satisfies ( φ ∞ ◦ ψ ∞ ) (cid:0) [[ x i ]] (cid:1) = [[ x i ]].Conversely, suppose that x := lim i → ω ψ i ( x i ) ∈ X ∞ exists. Fix any j ∈ N . For all i ∈ N sufficiently big, we have that (cid:12)(cid:12)(cid:12) d i (cid:0) ( φ i ◦ ψ j )( x j ) , x i (cid:1) − d ∞ (cid:0) ψ j ( x j ) , ψ i ( x i ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ ε i . In particular, it holds that d ω (cid:0) φ ∞ (cid:0) ψ j ( x j ) (cid:1) , [[ x i ]] (cid:1) = d ω (cid:0)(cid:2)(cid:2) ( φ i ◦ ψ j )( x j ) (cid:3)(cid:3) , [[ x i ]] (cid:1) = lim i → ω d i (cid:0) ( φ i ◦ ψ j )( x j ) , x i (cid:1) = lim i → ω d ∞ (cid:0) ψ j ( x j ) , ψ i ( x i ) (cid:1) = d ∞ (cid:0) ψ j ( x j ) , x (cid:1) . Consequently, by using the continuity of φ ∞ , we eventually conclude that d ω (cid:0) φ ∞ ( x ) , [[ x i ]] (cid:1) = lim j → ω d ω (cid:0) φ ∞ (cid:0) ψ j ( x j ) (cid:1) , [[ x i ]] (cid:1) = lim j → ω d ∞ (cid:0) ψ j ( x j ) , x (cid:1) = 0 . This grants that [[ x i ]] = φ ∞ ( x ) ∈ ψ ∞ ( X ∞ ), thus proving the first part of the statement.Now suppose ( X ∞ , d ∞ ) is proper. Fix [[ x i ]] ∈ X ω . Since lim i → ω d i ( x i , p i ) = d ω (cid:0) [[ x i ]] , p ω (cid:1) is finite, we can find R > S ∈ ω such that x i ∈ ¯ B ( p i , R ) for all i ∈ S . Moreover, thereexists j ∈ N such that ¯ B ( p i , R ) ⊂ B ( p i , R i ) and ε i ≤ i ≥ j , thus in particularwe have that ψ i ( x i ) ∈ ¯ B ( p ∞ , R + 1) for ω -a.e. i . Being ¯ B ( p ∞ , R + 1) compact, we concludethat the ultralimit lim i → ω ψ i ( x i ) ∈ X ∞ exists, whence it follows from the first part of thestatement that [[ x i ]] ∈ φ ∞ ( X ∞ ). This shows that X ω = φ ∞ ( X ∞ ), as desired. (cid:3) Theorem 9.4.
Let (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ ¯ N be a sequence of pointed Polish metric measure spaces,with m i boundedly finite. Suppose ( X i , d i , m i , p i ) → ( X ∞ , d ∞ , m ∞ , p ∞ ) in the pmGH sense.Then the measure m ω is concentrated on spt( m ω ) = φ ∞ (cid:0) spt( m ∞ ) (cid:1) and ( φ ∞ ) ∗ m ∞ = m ω . Proof.
We subdivide the proof into several steps:
Step 1.
Define Y := φ ∞ (cid:0) spt( m ∞ ) (cid:1) . Notice that Y is a Polish metric space if endowed withthe restricted distance from X ω . First of all, we aim to prove the inclusion spt( m ω ) ⊂ Y .Let [[ x i ]] ∈ spt( m ω ) be fixed. This means that ¯ m ω (cid:0) B (cid:0) [ x i ] , r (cid:1)(cid:1) = m ω (cid:0) B (cid:0) [[ x i ]] , r (cid:1)(cid:1) > r >
0. Thanks to (7.2), this is equivalent to saying thatlim i → ω m i (cid:0) B ( x i , r ) (cid:1) = ¯ m ω (cid:0) Π i → ω B ( x i , r ) (cid:1) > , for every r > . (9.2)Also, we have lim i → ω d i ( x i , p i ) = ¯ d ω (cid:0) [ x i ] , [ p i ] (cid:1) < + ∞ , so there are R > S ∈ ω such that x i ∈ B ( p i , R ) , for every i ∈ S. (9.3)Our aim is to show that [[ x i ]] ∈ Y . To achieve this goal, we first need to prove that ∃ x := lim i → ω ψ i ( x i ) ∈ X ∞ . (9.4)We argue by contradiction: suppose lim i → ω ψ i ( x i ) does not exist. By using Lemma 9.1, wecan find ε ∈ (0 ,
1) with the property that for every S ′ ∈ ω the set (cid:0) ψ i ( x i ) (cid:1) i ∈ S ′ cannot becovered by finitely many balls of radius ε . Call c := ¯ m ω (cid:0) Π i → ω B ( x i , ε/ (cid:1) / >
0. Let us define S := n i ∈ S (cid:12)(cid:12)(cid:12) d ∞ (cid:0) ψ i ( x i ) , ψ ( x ) (cid:1) ≥ ε, m i (cid:0) B ( x i , ε/ (cid:1) > c o and, for every k ∈ N , S k +1 := n i ∈ S k (cid:12)(cid:12)(cid:12) d ∞ (cid:0) ψ i ( x i ) , ψ i k ( x i k ) (cid:1) ≥ ε o , where we set i k := min( S k ) . Let us denote T k := (cid:8) i ∈ S : d ∞ (cid:0) ψ i ( x i ) , ψ i k ( x i k ) (cid:1) < ε (cid:9) for every k ∈ N . Given that itholds that (cid:0) ψ i ( x i ) (cid:1) i ∈ T k ⊂ B (cid:0) ψ i k ( x i k ) , ε (cid:1) , we deduce that T k / ∈ ω , so that N \ T k ∈ ω andaccordingly ( S k ) k ∈ N ⊂ ω . In particular, ( x i k ) k is a subsequence of ( x i ) i . Observe that d ∞ (cid:0) ψ i k ( x i k ) , ψ i j ( x i j ) (cid:1) ≥ ε, m i k (cid:0) B ( x i k , ε/ (cid:1) > c, (9.5)for every k, j ∈ N . Since P k ∈ N m ∞ (cid:0) B (cid:0) ψ i k ( x i k ) , ε/ (cid:1)(cid:1) ≤ m ∞ (cid:0) B ( p ∞ , R + 1) (cid:1) < + ∞ by (9.3),there exists ¯ k ∈ N such that m ∞ (cid:18) [ k ≥ ¯ k B (cid:0) ψ i k ( x i k ) , ε/ (cid:1)(cid:19) = X k ≥ ¯ k m ∞ (cid:0) B (cid:0) ψ i k ( x i k ) , ε/ (cid:1)(cid:1) ≤ c . (9.6)It can be readily checked that there exists a function f ∈ C bbs ( X ∞ ) with 0 ≤ f ≤ f = 1 on B (cid:0) ψ i k ( x i k ) , ε/ (cid:1) , for every k ≥ ¯ k,f = 0 on X ∞ \ S k ≥ ¯ k B (cid:0) ψ i k ( x i k ) , ε/ (cid:1) . LTRALIMITS OF POINTED METRIC MEASURE SPACES 35
Given that ψ i (cid:0) B ( x i , ε/ (cid:1) ⊂ B (cid:0) ψ i ( x i ) , ε/ (cid:1) for all i sufficiently big and ( ψ i ) ∗ m i ⇀ m ∞ weakly, c (9.5) ≤ lim k →∞ m i k (cid:0) B ( x i k , ε/ (cid:1) ≤ lim k →∞ m i k (cid:16) ψ − i k (cid:0) B (cid:0) ψ i k ( x i k ) , ε/ (cid:1)(cid:1)(cid:17) ≤ lim k →∞ ˆ f d( ψ i k ) ∗ m i k = ˆ f d m ∞ ≤ m ∞ (cid:16)S k ≥ ¯ k B (cid:0) ψ i k ( x i k ) , ε/ (cid:1)(cid:17) (9.6) ≤ c . This leads to a contradiction, thus accordingly the claim (9.4) is proven.
Step 2.
We now claim that x = lim i → ω ψ i ( x i ) ∈ spt( m ∞ ) . (9.7)Let r > λ := lim i → ω m i (cid:0) B ( x i , r/ (cid:1) >
0. By using thisfact, (9.3), and (9.4), we can find a subsequence ( i k ) k ∈ N such that for every k ∈ N it holds ε i k ≤ r , m i k (cid:0) B ( x i k , r/ (cid:1) ≥ λ , x i k ∈ B (cid:0) p i k , R i k − r/ (cid:1) , d ∞ (cid:0) ψ i k ( x i k ) , x (cid:1) < r . Given that B ( x i k , r/ ⊂ B ( p i k , R i k ) and the map ψ i k is a ( R i k , ε i k )-approximation, we havethat ψ i k (cid:0) B ( x i k , r/ (cid:1) ⊂ B (cid:0) ψ i k ( x i k ) , r/ ε i k (cid:1) ⊂ B (cid:0) ψ i k ( x i k ) , r/ (cid:1) , which in turn implies thatthe inclusion B ( x i k , r/ ⊂ ψ − i k (cid:0) B (cid:0) ψ i k ( x i k ) , r/ (cid:1)(cid:1) holds. Also, B (cid:0) ψ i k ( x i k ) , r/ (cid:1) ⊂ ¯ B ( x, r ) issatisfied for every k ∈ N . All in all, since ( ψ i k ) ∗ m i k ⇀ m ∞ weakly, we obtain that0 < λ ≤ lim k →∞ m i k (cid:0) B ( x i k , r/ (cid:1) ≤ lim k →∞ ( ψ i k ) ∗ m i k (cid:0) B (cid:0) ψ i k ( x i k ) , r/ (cid:1)(cid:1) ≤ lim k →∞ ( ψ i k ) ∗ m i k (cid:0) ¯ B ( x, r ) (cid:1) ≤ m ∞ (cid:0) ¯ B ( x, r ) (cid:1) . By arbitrariness of r >
0, we can conclude that x ∈ spt( m ∞ ), which yields (9.7). Finally, itfollows from (9.7) and Corollary 9.3 that [[ x i ]] ∈ Y . Hence, we showed that spt( m ω ) ⊂ Y . Step 3.
The next step is to prove that ( φ ∞ ) ∗ m ∞ ( B ) = m ω ( B ) is satisfied for every givenBorel set B ⊂ φ ∞ ( X ∞ ). By inner regularity of ( φ ∞ ) ∗ m ∞ and m ω , it is sufficient to show that m ω (cid:0) φ ∞ ( K ) (cid:1) = m ∞ ( K ) , for every K ⊂ X ∞ compact. (9.8)Fix any ε >
0. Then we can find 0 < δ < δ ′ < ε such that m ∞ ( ∂U ) = m ∞ ( ∂V ) = 0 and m ∞ ( V ) ≤ m ∞ ( K )+ ε , where U and V stand for the δ -neighbourhood and the δ ′ -neighbourhoodof K , respectively. Call C ε the closure of U . Note that K ⊂ U ⊂ C ε ⊂ V . We claim that π − (cid:0) φ ∞ ( X ∞ ) (cid:1) ∩ Π i → ω ψ − i ( U ) ⊂ π − (cid:0) φ ∞ ( C ε ) (cid:1) ⊂ Π i → ω ψ − i ( V ) . (9.9)First of all, let [ x i ] ∈ π − (cid:0) φ ∞ ( X ∞ ) (cid:1) ∩ Π i → ω ψ − i ( U ) be given. Consider the point x ∈ X ∞ for which (cid:2)(cid:2) φ i ( x ) (cid:3)(cid:3) = φ ∞ ( x ) = [[ x i ]]. We have that φ i ( x ) ∈ ψ − i ( U ) for ω -a.e. i , thus inparticular ( ψ i ◦ φ i )( x ) ∈ C ε for ω -a.e. i . Since lim i → ω d ∞ (cid:0) ( ψ i ◦ φ i )( x ) , x (cid:1) = 0 and C ε is closed,we conclude that x ∈ C ε as well. Therefore, we have that π (cid:0) [ x i ] (cid:1) = [[ x i ]] = φ ∞ ( x ) ∈ φ ∞ ( C ε ).Suppose [ y i ] ∈ π − (cid:0) φ ∞ ( C ε ) (cid:1) . Then there exists y ∈ C ε for which (cid:2)(cid:2) φ i ( y ) (cid:3)(cid:3) = φ ∞ ( y ) = [[ y i ]].Given that lim i → ω d ∞ (cid:0) ( ψ i ◦ φ i )( y ) , y (cid:1) = 0 and V is an open neighbourhood of y , we knowthat ψ i ( y i ) = ( ψ i ◦ φ i )( y ) ∈ V for ω -a.e. i . This means that [ y i ] ∈ Π i → ω ψ − i ( V ), as desired. Recall that m ω is concentrated on spt( m ω ) by Proposition 8.6 and that spt( m ω ) ⊂ φ ∞ ( X ∞ )by Steps i → ω ( ψ i ) ∗ m i ( U ) = ¯ m ω (cid:0) Π i → ω ψ − i ( U ) (cid:1) = ¯ m ω (cid:16) π − (cid:0) φ ∞ ( X ∞ ) (cid:1) ∩ Π i → ω ψ − i ( U ) (cid:17) ≤ m ω (cid:0) φ ∞ ( C ε ) (cid:1) ≤ ¯ m ω (cid:0) Π i → ω ψ − i ( V ) (cid:1) = lim i → ω ( ψ i ) ∗ m i ( V ) . (9.10)Given that U and V have m ∞ -negligible boundary, we know that lim i →∞ ( ψ i ) ∗ m i ( U ) = m ∞ ( U )and lim i →∞ ( ψ i ) ∗ m i ( V ) = m ∞ ( V ). Therefore, we deduce from (9.10) that m ω (cid:0) φ ∞ ( C ε ) (cid:1) ≤ lim i →∞ ( ψ i ) ∗ m i ( V ) = m ∞ ( V ) ≤ m ∞ ( U ) + ε = lim i →∞ ( ψ i ) ∗ m i ( U ) + ε ≤ m ω (cid:0) φ ∞ ( C ε ) (cid:1) + ε. This implies that (cid:12)(cid:12)(cid:12) m ω (cid:0) φ ∞ ( C ε ) (cid:1) − m ∞ ( C ε ) (cid:12)(cid:12)(cid:12) ≤ ε. (9.11)Now fix any sequence ε j ց
0. We can assume without loss of generality that C ε j +1 ⊂ C ε j forall j ∈ N . Since K = T j ∈ N C ε j by construction, we infer that φ ∞ ( K ) = T j ∈ N φ ∞ ( C ε j ), thusthe continuity from above of m ω and m ∞ gives (cid:12)(cid:12)(cid:12) m ω (cid:0) φ ∞ ( K ) (cid:1) − m ∞ ( K ) (cid:12)(cid:12)(cid:12) = lim j →∞ (cid:12)(cid:12)(cid:12) m ω (cid:0) φ ∞ ( C ε j ) (cid:1) − m ∞ ( C ε j ) (cid:12)(cid:12)(cid:12) (9.11) = 0 . This shows that m ω (cid:0) φ ∞ ( K ) (cid:1) = m ∞ ( K ), thus proving (9.8). Step 4.
In order to achieve the statement, it only remains to show that Y ⊂ spt( m ω ). Fixany point y ∈ Y . Take x ∈ spt( m ∞ ) with y = φ ∞ ( x ). Fix r >
0. Given that m ∞ (cid:0) B ( x, r ) (cid:1) > B ( x, r ) ⊂ φ − ∞ (cid:0) B ( y, r ) (cid:1) – the latter holds because φ ∞ is an isometry – we have that m ω (cid:0) B ( y, r ) (cid:1) = m ω (cid:0) B ( y, r ) ∩ Y (cid:1) = m ∞ (cid:0) φ − ∞ (cid:0) B ( y, r ) (cid:1) ∩ φ − ∞ ( Y ) (cid:1) = m ∞ (cid:0) φ − ∞ (cid:0) B ( y, r ) (cid:1) ∩ spt( m ∞ ) (cid:1) = m ∞ (cid:0) φ − ∞ (cid:0) B ( y, r ) (cid:1)(cid:1) ≥ m ∞ (cid:0) B ( x, r ) (cid:1) > . By arbitrariness of r >
0, we conclude that y ∈ spt( m ω ), proving the inclusion Y ⊂ spt( m ω ).Consequently, the proof is complete. (cid:3) HAPTER IV
Weak pointed measured Gromov–Hausdorff convergence
Definition of wpmGH convergence
In this section, we will give a definition of a weaker version of pointed measured Gromov–Hausdorff convergence, and investigate its connection with the notion of ultralimit of metricmeasure spaces. We will generalise Gromov’s compactness theorem to this setting, and withthat, we will show that this new notion of convergence is in fact equivalent to the notion ofthe so-called pointed measured Gromov convergence, introduced and studied in [21].
Definition 10.1 (Weak (
R, ε )-approximation) . Let ( X, d X , m X , p X ) and ( Y, d Y , m Y , p Y ) bepointed Polish metric measure spaces. Let R, ε > be such that ε < R . Then a givenBorel map ψ : B ( p X , R ) → Y is said to be a weak ( R, ε )-approximation provided it satisfies ψ ( p X ) = p Y , and there exists a Borel set ˜ X ⊂ B ( p X , R ) containing p X such that (1) d X ( x, y ) − ε ≤ d Y (cid:0) ψ ( x ) , ψ ( y ) (cid:1) ≤ d X ( x, y ) + ε for every x, y ∈ ˜ X , (2) m X (cid:0) B ( p X , R ) \ ˜ X (cid:1) ≤ ε and m Y (cid:0) B ( p Y , R − ε ) \ ψ ( ˜ X ) ε (cid:1) ≤ ε . Remark 4.2 applies also here: the map ψ can be viewed as a map defined globally on thewhole space X . Notice that the definition is weaker than but analogous to Definition 4.1;the condition (1) says that the map ψ is (1 , ε )-quasi-isometric in the ball of radius R afterneglecting a small set in measure, while the condition (2) states that the map is roughlysurjective up to a small error in measure.The following lemma grants that also in the setting of weak ( R, ε )-approximations, we havethe existence of rough inverses. Its proof is very similar to the one of Lemma 4.3; we sketchthe argument for completeness.
Lemma 10.2.
Let ( X, d X , m X , p X ) , ( Y, d Y , m Y , p Y ) be pointed Polish metric measure spaces.Let ψ : X → Y be a given weak ( R, ε ) -approximation, for some R, ε > satisfying ε < R .Then there exists a weak ( R − ε, ε ) -approximation φ : Y → X – that we will call a roughinverse of ψ – such that d X (cid:0) x, ( φ ◦ ψ )( x ) (cid:1) < ε, for every x ∈ B ( p X , R − ε ) ∩ ˜ X,d Y (cid:0) y, ( ψ ◦ φ )( y ) (cid:1) < ε, for every y ∈ B ( p Y , R − ε ) ∩ ˜ Y .
Proof.
Define ˜ Y := ψ ( ˜ X ) ε ∩ B ( p Y , R − ε ). Let { x i } i ∈ N be dense in ˜ X . Given any y ∈ ˜ Y \{ p Y } ,define φ ( y ) := x i y , where i y is the smallest index i ∈ N for which d Y (cid:0) ψ ( x i ) , y (cid:1) < ε . We alsodefine φ ( p Y ) := p X , while we set φ ( y ) := x for all y ∈ Y \ ˜ Y , where x ∈ X \ B ( p X , R ) is any
378 ENRICO PASQUALETTO AND TIMO SCHULTZ given point (when X \ B ( p X , R ) = ∅ , otherwise x := p X ). The resulting map φ : Y → X isBorel and satisfies φ ( p Y ) = p X by construction. For every y, y ′ ∈ ˜ Y , we have that (cid:12)(cid:12)(cid:12) d X (cid:0) φ ( y ) , φ ( y ′ ) (cid:1) − d Y ( y, y ′ ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) d X (cid:0) φ ( y ) , φ ( y ′ ) (cid:1) − d Y (cid:0) ( ψ ◦ φ )( y ) , ( ψ ◦ φ )( y ′ ) (cid:1)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) d Y (cid:0) ( ψ ◦ φ )( y ) , ( ψ ◦ φ )( y ′ ) (cid:1) − d Y ( y, y ′ ) (cid:12)(cid:12)(cid:12) ≤ ε + d Y (cid:0) ( ψ ◦ φ )( y ) , y (cid:1) + d Y (cid:0) ( ψ ◦ φ )( y ′ ) , y ′ (cid:1) < ε, which shows that sup y,y ′ ∈ ˜ Y (cid:12)(cid:12) d X (cid:0) φ ( y ) , φ ( y ′ ) (cid:1) − d Y ( y, y ′ ) (cid:12)(cid:12) ≤ ε . Moreover, given any point x in B ( p X , R − ε ) ∩ ˜ X , we have that ψ ( x ) ∈ B ( p Y , R − ε ) ∩ ˜ Y , thus from the validity of theinequalities d X (cid:0) ( φ ◦ ψ )( x ) , x (cid:1) ≤ d Y (cid:0) ( ψ ◦ φ ◦ ψ )( x ) , ψ ( x ) (cid:1) + ε < ε < ε it follows that x belongs to the 3 ε -neighbourhood of φ (cid:0) B ( p Y , R − ε ) ∩ ˜ Y (cid:1) . In particular, itholds that B ( p X , R − ε ) \ φ ( ˜ Y ) ε ⊂ B ( p X , R − ε ) \ ˜ X , hence m X (cid:0) B ( p X , R − ε ) \ φ ( ˜ Y ) ε (cid:1) ≤ m X (cid:0) B ( p X , R − ε ) \ ˜ X (cid:1) ≤ ε. Accordingly, φ is a weak ( R − ε, ε )-approximation. In order to conclude, it only remains toobserve that (4.1b) is a direct consequence of the very definition of φ , while (4.1a) followsfrom the estimate in (4.2). (cid:3) Definition 10.3 (Weak pointed measured Gromov–Hausdorff convergence) . Let us considera sequence (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ ¯ N of pointed Polish metric measure spaces. Then ( X i , d i , m i , p i ) is said to converge to ( X ∞ , d ∞ , m ∞ , p ∞ ) in the weak pointed measured Gromov–Hausdorffsense (briefly, in the wpmGH sense ) as i → ∞ provided there exists a sequence ( ψ i ) i ∈ N ofweak ( R i , ε i ) -approximations, for some R i ր + ∞ and ε i ց , so that ( ψ i ) ∗ m i ⇀ m ∞ , in duality with C bbs ( X ∞ ) . Relation between wpmGH convergence and ultralimits
In the following proof of Gromov compactness theorem for weak pointed measured Gromov–Hausdorff convergence, we will need the variant of Prokhorov theorem for ultralimits intro-duced in Appendix A.
Theorem 11.1.
Let (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ N be a given asymptotically uniformly boundedly m ω -totally bounded sequence of pointed Polish metric measure spaces. Then it holds that theultralimit lim i → ω ( X i , d i , m i , p i ) with respect to the wpmGH-topology exists and is isomorphicto (cid:0) spt ( m ω ) , d ω , m ω , p ω (cid:1) . Remark 11.2.
Technically speaking, we have given the definition of an ultralimit of pointsonly when the underlying space is a (pseudo)metric space. The ultralimit in the above theoremcan be regarded as such after proving the equivalence of wpmGH- and pmG-convergence inTheorem 12.2, and noticing that the pmG-topology is in fact metrisable.For the reader who is not comfortable with such an undirect reasoning, we suggest to thinkof Theorem 11.1 more directly in terms of weak (
R, ε )-approximations: it simply states that
LTRALIMITS OF POINTED METRIC MEASURE SPACES 39 there exist sequences ˜ R i , ˜ ε i > R i , ˜ ε i )-approximations ψ i from X i to spt ( m ω ) sothat lim i → ω R i = ∞ , lim i → ω ε i = 0, andlim i → ω ( ψ i ) ∗ m i = m ω . Here, the last ultralimit makes sense, since the weak convergence is metrisable. (cid:4)
Proof of Theorem 11.1.
The proof goes along the same lines as the proof of the classicalGromov’s compactness theorem using ultralimits. We subdivide the proof into several steps:
Step 1.
Let ( X i , d i , m i , p i ) be an asymptotically uniformly boundedly m ω -totally boundedsequence of pointed Polish metric measure spaces. Let R i ր ∞ and ε i ց Y i ⊂ B ( p ω , R i ) ∩ spt ( m ω ) be a compact set, with ˜ Y i ⊂ ˜ Y i +1 , and let N i ∈ N be so that m ω (cid:0) B ( p ω , R i ) \ ˜ Y i (cid:1) ≤ ε i (11.1)and ˜ Y i ⊂ N i [ j =1 B ( x i,j , ε i ) (11.2)for some ( x i,j ) N i j =1 ⊂ ˜ Y i such that d ω ( x i,j , x i,m ) > ε i whenever j, m = 1 , . . . , N i satisfy j = m .Furthermore, we can additionally require that for any i < i ′ it holds that N i ≤ N i ′ and x i ′ ,j = x i,j for every j = 1 , . . . , N i . For each i, j , we can write x i,j = [[ x i,jk ]] k , where x i,jk ∈ X k for all k ∈ N . Observe that we can further require that x i,jk = x i ′ ,jk for every i < i ′ , j = 1 , . . . , N i , and k ∈ N .We now define for each i ∈ N the sets F i recursively by setting F := A ∩ B ∩ C ∩ D and F i +1 := F i ∩ A i +1 ∩ B i +1 ∩ C i +1 ∩ D i +1 , where we define A i := n k ≥ i (cid:12)(cid:12)(cid:12) m k (cid:0) B ( p k , R i ) \ S N i j =1 B ( x i,jk , ε i ) (cid:1) ≤ ε i o , B i := n k ∈ N (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) d k ( x i,jk , x i,mk ) − d ω ( x i,j , x i,m ) (cid:12)(cid:12) < ε i for all j, m ≤ N i o , C i := n k ∈ N (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) d k ( x i,jk , p k ) − d ω ( x i,j , p ω ) (cid:12)(cid:12) < ε i for all j ≤ N i (cid:9) , D i := n k ∈ N (cid:12)(cid:12)(cid:12) d k ( x i,jk , x i,mk ) > ε i for all j, m ≤ N i with j = m o . By asymptotic uniform bounded m ω -total boundedness (and by definition of ultralimit), wehave that F i ∈ ω holds for every i ∈ N . For every k ∈ F , there exists a unique i k ∈ N so that k ∈ F i k \ F i k +1 . For k ∈ F , define the Borel map ψ k : X k → X ω as follows: ψ k ( p k ) := p ω , ψ k ( x ) := x i k , , for every x ∈ B ( x i k , k , ε i k ) ,ψ k ( x ) := x i k ,j +1 , for every x ∈ B ( x i k ,j +1 k , ε i k ) \ S jn =1 B ( x i k ,jk , ε i k ) , while we can assume without loss of generality that ψ k (cid:0) X k \ S N ik j =1 B ( x i k ,jk , ε i k ) (cid:1) ⊂ X ω \ B ( p ω , R i k ) , (11.3) provided the latter set is not empty. Put ˜ R k := R i k and ˜ ε k := 6 ε i k for all k ∈ N . Then wehave (by Lemma 5.4) that lim k → ω ˜ R k = ∞ and lim k → ω ˜ ε k = 0. Step 2.
We claim that ψ k is a weak ( ˜ R k , ˜ ε k )-approximation, for every k ∈ F . In order to prove condition (1) of Definition 10.1, define˜ X k := N ik [ j =1 B ( x i k ,jk , ε i k ) . Let the points x, y ∈ ˜ X k be fixed. Then we have that x ∈ B ( x i k ,j x +1 k , ε i k ) \ S j x j =1 B ( x i k ,jk , ε i k )and y ∈ B ( x i k ,j y +1 k , ε i k ) \ S j y j =1 B ( x i k ,jk , ε i k ) for some j x , j y ≤ N i k −
1. If j x = j y , we have d ω (cid:0) ψ k ( x ) , ψ k ( y ) (cid:1) − ˜ ε k = − ε i k ≤ d k ( x, y ) ≤ ε i k ≤ d ω (cid:0) ψ k ( x ) , ψ k ( y ) (cid:1) + ˜ ε k . If j x = j y , we have d ω (cid:0) ψ k ( x ) , ψ k ( y ) (cid:1) − ˜ ε k = d ω ( x i k ,j x +1 , x i k ,j y +1 ) − ε i k ≤ d k ( x i k ,j x +1 k , x i k ,j y +1 k ) − ε i k ≤ d k ( x i k ,j x +1 k , x ) + d k ( x, y ) + d k ( y, x i k ,j y +1 k ) − ε i k ≤ d k ( x, y ) ≤ d k ( x i k ,j x +1 k , x i k ,j y +1 k ) + 4 ε i k ≤ d ω (cid:0) ψ k ( x ) , ψ k ( y ) (cid:1) + 5 ε i k ≤ d ω (cid:0) ψ k ( x ) , ψ k ( y ) (cid:1) + ˜ ε k . Hence, the property (1) is proven. For the property (2), first notice that the inequality m k (cid:0) B ( p k , ˜ R k ) \ ˜ X k (cid:1) ≤ ˜ ε k is true due to the definitions of ˜ X k and F i k . Moreover, given that { x i k ,j } N ik j =1 ⊂ ψ k ( ˜ X k ) bythe very construction of ψ k , we deduce from (11.2) that ˜ Y i k ⊂ S N ik j =1 B ( x i k ,j , ˜ ε k ) ⊂ ψ k ( ˜ X k ) ˜ ε k ,whence it follows from (11.1) that m ω (cid:0) B ( p ω , ˜ R k ) \ ψ k ( ˜ X k ) ˜ ε k (cid:1) ≤ ˜ ε k , yielding (2). All in all, wehave proven that ψ k is a weak ( ˜ R k , ˜ ε k )-approximation for every k ∈ F , i.e. , the claim (11.3). Step 3.
Next, we claim that there exists a Borel measure ˜ m ≥ m ω ) such that˜ m = lim k → ω ( ψ k ) ∗ m k , in duality with C bbs (cid:0) spt ( m ω ) (cid:1) . (11.4)Let n ∈ N and ε > k ∈ N such that ˜ R k ≥ ˜ R k , ˜ R k − ˜ ε k ≥ n , and 2˜ ε k ≤ ε for all k ∈ F i k with k ≥ k . Call µ nk := (cid:0) ( ψ k ) ∗ m k (cid:1) x (cid:0) B ( p ω , n ) ∩ spt ( m ω ) (cid:1) for every k ∈ F i k with k ≥ k and T := B ( ˜ Y i k , ε i k ). Since ˜ Y i k is compact, one has that T is 4 ε i k -totallybounded, thus in particular it is ε -totally bounded. Moreover, we claim that ψ k (cid:18) N ik [ j =1 B (cid:0) x i k ,jk , ε i k (cid:1)(cid:19) ⊂ T, for every k ∈ F i k with k ≥ k . (11.5)In order to prove it, fix any j = 1 , . . . , N i k and x ∈ ˜ X k ∩ B (cid:0) x i k ,jk , ε i k (cid:1) = B (cid:0) x i k ,jk , ε i k (cid:1) .Being the map ψ k a weak ( ˜ R k , ˜ ε k )-approximation, we deduce that d ω (cid:0) ψ k ( x ) , ψ k ( x i k ,jk ) (cid:1) ≤ d k ( x, x i k ,jk ) + ˜ ε k < ε i k + ˜ ε k ≤ ε k . LTRALIMITS OF POINTED METRIC MEASURE SPACES 41
Given that ψ k ( x i k ,jk ) = ψ k ( x i k ,jk ) = x i k ,j = x i k ,j ∈ ˜ Y i k , we have proven that ψ k ( x ) ∈ T ,whence (11.5) follows. Therefore, since for any k ∈ F i k with k ≥ k it holds that ψ − k (cid:16)(cid:0) B ( p ω , n ) ∩ spt ( m ω ) (cid:1) \ T (cid:17) (11.5) ⊂ ψ − k (cid:0) B ( p ω , ˜ R k − ˜ ε k ) (cid:1) \ N ik [ j =1 B (cid:0) x i k ,jk , ε i k (cid:1) ⊂ B ( p k , ˜ R k ) \ N ik [ j =1 B (cid:0) x i k ,jk , ε i k (cid:1) , we conclude that µ nk (cid:0) spt ( m ω ) \ T (cid:1) = m k (cid:16) ψ − k (cid:16)(cid:0) B ( p ω , n ) ∩ spt ( m ω ) (cid:1) \ T (cid:17)(cid:17) ≤ m k (cid:16) B ( p k , ˜ R k ) \ N ik [ j =1 B (cid:0) x i k ,jk , ε i k (cid:1)(cid:17) ≤ ε i k ≤ ε. In particular, µ nk (cid:0) spt ( m ω ) \ T (cid:1) ≤ ε holds for ω -a.e. k . Hence, by using Theorem A.5 we obtainthat there exists a finite Borel measure ˜ m n on spt ( m ω ) for which ˜ m n = lim k → ω µ nk with respectto the weak convergence. It can be readily checked that ˜ m n = ˜ m n ′ x (cid:0) B ( p ω , n ) ∩ spt ( m ω ) (cid:1) whenever n < n ′ , so that it makes sense to define the measure ˜ m on spt ( m ω ) as˜ m ( E ) := lim n →∞ ˜ m n (cid:0) E ∩ B ( p ω , n ) (cid:1) , for every E ∈ B (cid:0) spt ( m ω ) (cid:1) . Finally, given any f ∈ C bbs (cid:0) spt ( m ω ) (cid:1) and n ∈ N with spt ( f ) ⊂ B ( p ω , n ), we have that ˆ f d ˜ m = ˆ f d ˜ m n = lim k → ω ˆ f d µ nk = lim k → ω ˆ f d( ψ k ) ∗ m k . Therefore, we have proven that (11.4) is satisfied, as desired.
Step 4.
In order to conclude, it only remains to show that there exists an isometric bijection φ ∞ : spt ( m ω ) → spt ( m ω ) such that ( φ ∞ ) ∗ ˜ m = m ω | B (spt ( m ω )) . Since the arguments are verysimilar to those in the proofs of the results in Section 9, we will omit some details.First of all, we define φ ∞ : spt ( m ω ) → spt ( m ω ). Given any y ∈ spt ( m ω ), there exist y i ∈ ˜ Y i so that y = lim i → ω y i . Then let us define φ ∞ ( y ) := (cid:2)(cid:2) φ i ( y i ) (cid:3)(cid:3) i ∈ X ω , where φ i is a roughinverse of ψ i (whose existence is granted by Lemma 10.2). The map φ ∞ is well-defined andisometric by Definition 10.3 of wpmGH-convergence and by (2) of Definition 10.1. Moreover,given any point x = φ ∞ ( y ) ∈ φ ∞ (cid:0) spt ( m ω ) (cid:1) , we define ψ ∞ ( x ) := lim i → ω ψ i ( x i ) ∈ spt ( m ω ) , where x = [[ x i ]] i for some x i ∈ ˜ X i . Then ψ ∞ is well-defined and is the inverse of ψ ∞ , sincelim i → ω d ω (cid:0) y, ψ i ( x i ) (cid:1) ≤ lim i → ω h d ω ( y, y i ) + d ω (cid:0) y i , ( ψ i ◦ φ i )( y i ) (cid:1) + d ω (cid:0) ( ψ i ◦ φ i )( y i ) , ψ i ( x i ) (cid:1)i = 0 . As in
Step 1 of the proof of Theorem 9.4, we get spt ( m ω ) ⊂ φ ∞ (cid:0) spt ( ˜ m ) (cid:1) ⊂ φ ∞ (cid:0) spt ( m ω ) (cid:1) .We will show that m ω x φ ∞ (cid:0) spt ( m ω ) (cid:1) = ( φ ∞ ) ∗ ˜ m . Since ˜ m and m ω are both boundedly finite, it suffices to prove that ˜ m ( K ) = m ω (cid:0) φ ∞ ( K ) (cid:1) for every K ⊂ spt ( m ω ) compact. Let K ⊂ spt ( m ω )be a compact set and ε >
0. Then there exist 0 < δ < δ ′ < ε so that ˜ m ( ∂U ) = ˜ m ( ∂V ) = 0 and˜ m ( V ) ≤ ˜ m ( K ) + ε , where U and V are the δ - and the δ ′ -neighbourhood of K , respectively.Denote by C ε the closure of U . Notice that K ⊂ U ⊂ C ε ⊂ V . Let us show that π − (cid:0) φ ∞ (cid:0) spt ( m ω ) (cid:1)(cid:1) ∩ Π i → ω ψ − i ( U ) ⊂ π − (cid:0) φ ∞ ( C ε ) (cid:1) ⊂ Π i → ω ψ − i ( V ) . (11.6)Let [ x i ] ∈ Π i → ω ψ − i ( U ) be so that [[ x i ]] = φ ∞ ( y ) for some y ∈ spt ( m ω ). Then φ i ( y i ) ∈ ψ − i ( U )for ω -almost every i ∈ N , where φ ∞ ( y ) = (cid:2)(cid:2) φ i ( y i ) (cid:3)(cid:3) . Thus, ( ψ i ◦ φ i )( y i ) ∈ C ε for ω -almostevery i ∈ N , and therefore, since C ε is closed, we have that y = lim i → ω ψ i ( x i ) = lim i → ω ( ψ i ◦ φ i )( y i ) ∈ C ε . Thus, we have proven that [ x i ] ∈ π − (cid:0) φ ∞ ( C ε ) (cid:1) . Now suppose that [ z i ] ∈ π − (cid:0) φ ∞ ( C ε ) (cid:1) . Thenthere exists z ∈ C ε for which (cid:2)(cid:2) φ i ( z ) (cid:3)(cid:3) = φ ∞ ( z ) = [[ z i ]]. Since lim i → ω d ∞ (cid:0) ( ψ i ◦ φ i )( z ) , z (cid:1) = 0and V is an open neighbourhood of z , we know that ψ i ( z i ) = ( ψ i ◦ φ i )( z ) ∈ V for ω -a.e. i ∈ N .This means that [ y i ] ∈ Π i → ω ψ − i ( V ), as desired. All in all, the claim (11.6) is proven.Since m ω is concentrated on spt ( m ω ), and spt ( m ω ) ⊂ φ ∞ (cid:0) spt ( m ω ) (cid:1) , we have by (11.6)thatlim i → ω ( ψ i ) ∗ m i ( U ) = ¯ m ω (cid:0) Π i → ω ψ − ( U ) (cid:1) = ¯ m ω (cid:0) π − (cid:0) φ ∞ (spt ( m ω )) (cid:1) ∩ Π i → ω ψ − ( U ) (cid:1) ≤ m ω (cid:0) ψ ∞ ( C ε ) (cid:1) ≤ ¯ m ω (cid:0) Π i → ω ψ − i ( V ) (cid:1) = lim i → ω ( ψ i ) ∗ m i ( V ) . (11.7)Due to the fact that U and V are continuity sets for ˜ m , we obtain from (11.7) that m ω (cid:0) φ ∞ ( C ε ) (cid:1) ≤ lim i → ω ( ψ i ) ∗ m i ( V ) = ˜ m ( V ) ≤ ˜ m ( K ) + ε ≤ ˜ m ( U ) + ε = lim i → ω ( ψ i ) ∗ m i ( U ) + ε ≤ m ω (cid:0) φ ∞ ( C ε ) (cid:1) + ε. By using the continuity from above of m ω and letting ε →
0, we obtain ˜ m ( K ) = m ω (cid:0) φ ∞ ( K ) (cid:1) .Finally, we conclude by observing that, arguing as in Step 4 of the proof of Theorem 9.4,we get that φ ∞ (cid:0) spt ( ˜ m ) (cid:1) ⊂ spt ( m ω ). Consequently, the statement is achieved. (cid:3) Corollary 11.3.
Let (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ N be a sequence of pointed Polish metric measurespaces that wpmGH-converges to a limit space ( X ∞ , d ∞ , m ∞ , p ∞ ) . Then (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ N is asymptotically boundedly m ω -totally bounded and its wpmGH-limit ( X ∞ , d ∞ , m ∞ , p ∞ ) isisomorphic to (cid:0) spt ( m ω ) , d ω , m ω , p ω (cid:1) .Proof. Thanks to Theorem 11.1, it suffices to prove that the sequence (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ N is asymptotically boundedly m ω -totally bounded. This can be easily achieved by suitablyadapting the arguments in the proof of Proposition 8.6; we omit the details. (cid:3) Before stating the main result of this section, we need to introduce a couple of definitions.Let (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ N be a sequence of pointed Polish metric measure spaces. Then we saythat (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ N is uniformly boundedly finite provided for any R > i ∈ N m i (cid:0) B ( p i , R ) (cid:1) < + ∞ . LTRALIMITS OF POINTED METRIC MEASURE SPACES 43
In addition, we say that (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ N is boundedly measure-theoretically totally bounded provided for every R, r, ε > M ∈ N and points ( x in ) Mn =1 ⊂ X i such thatsup i ∈ N m i (cid:0) ¯ B ( p i , R ) \ S Mn =1 B ( x in , r ) (cid:1) ≤ ε. Theorem 11.4 (Gromov’s compactness for wpmGH-convergence) . Let (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ N be a sequence of boundedly finite pointed Polish metric measure spaces. Then it holds that (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ N is precompact with respect to the wpmGH-convergence ( i.e. , any of itssubsequences has a wpmGH-converging subsequence) if and only if it is uniformly boundedlyfinite and boundedly measure-theoretically totally bounded. Remark 11.5.
We do not claim that the limit point p ∞ is in the support of the limit measurenor that the limit measure is non-trivial. To guarantee that, one can obtain another compactclass of metric measure spaces by requiring also that m i (cid:0) B ( p i , R ) (cid:1) is uniformly bounded awayfrom 0 for every R > (cid:4) Proof of Theorem 11.4.
Necessity.
Suppose (cid:8) ( X i , d i , m i , p i ) (cid:9) i is wpmGH-precompact. We argue by contradiction:suppose that there exists a subsequence (cid:8) ( X i j , d i j , m i j , p i j ) (cid:9) j that does not admit any bound-edly measure-theoretically totally bounded subsequence.Now we can pick any wpmGH-converging subsequence (cid:8) ( X i jk , d i jk , m i jk , p i jk ) (cid:9) k , which isasymptotically boundedly m ω -totally bounded by Corollary 11.3 and accordingly has a bound-edly measure-theoretically totally bounded subsequence. This leads to a contradiction, thus (cid:8) ( X i , d i , m i , p i ) (cid:9) i is a boundedly measure-theoretically totally bounded sequence, as desired. Sufficiency.
Suppose the sequence (cid:8) ( X i , d i , m i , p i ) (cid:9) i is boundedly measure-theoreticallytotally bounded. Fix an arbitrary subsequence (cid:8) ( X i j , d i j , m i j , p i j ) (cid:9) j , which is boundedlymeasure-theoretically totally bounded as well. In particular, it is asymptotically boundedly m ω -totally bounded, thus it has a wpmGH-converging subsequence by Theorem 11.1. Thisshows that (cid:8) ( X i , d i , m i , p i ) (cid:9) i is wpmGH-precompact, yielding the sought conclusion. (cid:3) Equivalence between wpmGH convergence and pmG convergence
We will use Theorem 11.4 to prove that the weak pointed measured Gromov–Hausdorffconvergence is actually equivalent to the so-called pointed measured Gromov convergence (briefly, pmG convergence ), which was introduced in [21]. Amongst the several equivalentways to define the pmG convergence, we just need to recall the ‘extrinsic approach’ [21,Definition 3.9].
Definition 12.1 (Pointed measured Gromov convergence) . Let (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ ¯ N be a givensequence of pointed Polish metric measure spaces. Then ( X i , d i , m i , p i ) is said to convergeto ( X ∞ , d ∞ , m ∞ , p ∞ ) in the pointed measured Gromov sense (briefly, in the pmG sense )provided there exist a Polish metric space ( Y, d ) and isometric embeddings ι i : X i → Y , i ∈ ¯ N ,such that lim i →∞ d (cid:0) ι i ( p i ) , ι ∞ ( p ∞ ) (cid:1) = 0 and ( ι i ) ∗ m i ⇀ ( ι ∞ ) ∗ m ∞ , in duality with C bbs ( Y ) . Theorem 12.2 (wpmGH and pmG) . Let (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ ¯ N be given pointed Polish metricmeasure spaces. Then it holds that ( X i , d i , m i , p i ) −→ ( X ∞ , d ∞ , m ∞ , p ∞ ) , in the wpmGH-senseif and only if ( X i , d i , m i , p i ) −→ ( X ∞ , d ∞ , m ∞ , p ∞ ) , in the pmG-sense.Proof. Suppose ( X i , d i , m i , p i ) is wpmGH-converging to ( X ∞ , d ∞ , m ∞ , p ∞ ). We aim to showthat ( X i , d i , m i , p i ) → ( X ∞ , d ∞ , m ∞ , p ∞ ) with respect to the pmG-topology. The proof goesalong the same lines as the proof of [21, Proposition 3.30]. Let us define a distance d on theset Y := F i ∈ ¯ N X i so that the spaces X i ’s are isometrically embedded into Y and m i ⇀ m ∞ weakly. Let ψ i : X i → X ∞ be the weak ( R i , ε i )-approximations given by the definition ofwpmGH-convergence. Then we define d : Y × Y → [0 , + ∞ ) as d ( y , y ) := d i ( y , y ) , Φ i ( y , y ) , Ψ ij ( y , y ) , if y , y ∈ X i for some i ∈ ¯ N , if y ∈ X i for some i ∈ N and y ∈ X ∞ , if y ∈ X i and y ∈ X j for some i, j ∈ N with i = j, where the functions Φ i : X i × X ∞ → [0 , + ∞ ) and Ψ ij : X i × X j → [0 , + ∞ ) are given byΦ i ( y , y ) := inf n d i ( x i , y ) + d ∞ (cid:0) ψ i ( x i ) , y (cid:1) + ε i (cid:12)(cid:12)(cid:12) x i ∈ ˜ X i o , Ψ ij ( y , y ) := inf n d i ( x i , y ) + d j ( x j , y ) + d ∞ (cid:0) ψ i ( x i ) , ψ j ( x j ) (cid:1) + ε i + ε j (cid:12)(cid:12)(cid:12) x i ∈ ˜ X i , x j ∈ ˜ X j o . Standard verifications show that d is a distance on Y . Note that for any x ∈ ˜ X i one has d (cid:0) x, ψ i ( x ) (cid:1) ≤ ε i . (12.1)In particular, d ( p i , p ∞ ) → i → ∞ . By weak convergence of ( ψ i ) ∗ m i , we have that η ( R ) := lim sup i →∞ m i (cid:0) B ( p i , R ) (cid:1) < + ∞ , for every R > . We need to show that m i ⇀ m ∞ weakly in duality with C bbs ( Y ). First, we claim that ˆ f d m i −→ ˆ f d m ∞ , for every f ∈ C bbs ( Y ) Lipschitz. (12.2)Let any such f be fixed, say that f is L -Lipschitz. Pick R > f = 0 on Y \ B ( p i , R/ i . Since ( ψ i ) ∗ m i ⇀ m ∞ , we have ´ f ◦ ψ i d m i → ´ f d m ∞ . Thus, it suffices to prove (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d m i − ˆ f ◦ ψ i d m ∞ (cid:12)(cid:12)(cid:12)(cid:12) −→ , as i → ∞ . By (12.1), we have (for every i with R i ≥ R ) that (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d m i − ˆ f ◦ ψ i d m ∞ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ ˜ X i ∩ B ( p i ,R ) f d m i − ˆ ˜ X i ∩ B ( p i ,R ) f ◦ ψ i d m ∞ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ˆ B ( p i ,R i ) \ ˜ X i f d m i − ˆ X i \ ˜ X i f ◦ ψ i d m ∞ (cid:12)(cid:12)(cid:12)(cid:12) ≤ L ε i m i (cid:0) B ( p i , R ) (cid:1) + 2 ε i sup Y | f | , LTRALIMITS OF POINTED METRIC MEASURE SPACES 45 whence the claimed property (12.2) follows. Now, fix any f ∈ C bbs ( Y ) and ε >
0. Chooseany function ˜ f ∈ C bbs ( Y ) Lipschitz with sup Y | f − ˜ f | ≤ ε . For R > i →∞ (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d m i − ˆ f d m ∞ (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim i →∞ ˆ | f − ˜ f | d m i + lim i →∞ (cid:12)(cid:12)(cid:12)(cid:12) ˆ ˜ f d m i − ˆ ˜ f d m ∞ (cid:12)(cid:12)(cid:12)(cid:12) + ˆ | ˜ f − f | d m ∞ (12.2) ≤ (cid:16) η ( R ) + m ∞ (cid:0) B ( p ∞ , R ) (cid:1)(cid:17) ε. By the arbitrariness of ε >
0, we deduce that ´ f d m ∞ = lim i →∞ ´ f d m i . This proves that m i ⇀ m ∞ in duality with C bbs ( Y ), so ( X i , d i , m i , p i ) → ( X ∞ , d ∞ , m ∞ , p ∞ ) in the pmG-sense.To prove the converse implication, suppose that ( X i , d i , m i , p i ) → ( X ∞ , d ∞ , m ∞ , p ∞ ) inthe pmG-sense. By Gromov’s compactness for pmG-convergence [21, Corollary 3.22] andby Theorems 11.1 and 11.4, we have that (up to passing to a not relabeled subsequence)( X i , d i , m i , p i ) → (cid:0) spt ( m ω ) , d ω , m ω , p ω (cid:1) in the wpmGH-sense. Then by using the first part ofthe proof we obtain that ( X i , d i , m i , p i ) → (cid:0) spt ( m ω ) , d ω , m ω , p ω (cid:1) in the pmG-topology. Byuniqueness of the limit, we conclude. Therefore, the statement is finally achieved. (cid:3) HAPTER V
Direct and inverse limits of pointed metric measure spaces
The category of pointed metric measure spaces
In this section, when we consider a pointed Polish metric measure space (
X, d, m , p ), weimplicitly assume that p ∈ spt( m ). Definition 13.1.
Let ( X, d X , m X , p X ) and ( Y, d Y , m Y , p Y ) be pointed Polish metric measurespaces. Then a map ϕ : X → Y is said to be a morphism of pointed Polish metric measurespaces if: i) The map ϕ is Borel measurable and satisfies ϕ ( p X ) = p Y . ii) The restricted map ϕ | spt( m X ) : spt( m X ) → Y is -Lipschitz. iii) It holds that ϕ ∗ m X ≤ m Y . With the above notion of morphism at our disposal, we can consider the category ofpointed Polish metric measure spaces. Observe that a morphism ϕ from ( X, d X , m X , p X )to ( Y, d Y , m Y , p Y ) is an isomorphism if and only if ϕ | spt( m X ) : spt( m X ) → Y is an isometryand ϕ ∗ m X = m Y . Below, we briefly remind the notions of direct and inverse limits of a se-quence of pointed Polish metric measure spaces, referring to the monograph [29] for a generaltreatment of these topics.A couple (cid:0) { ( X i , d i , m i , p i ) } i ∈ N , { ϕ ij } i ≤ j (cid:1) is said to be a direct system of pointed Polishmetric measure spaces provided each ( X i , d i , m i , p i ) is a pointed Polish metric measure spaceand the maps ϕ ij : X i → X j are morphisms of pointed Polish metric measure spaces satisfying ϕ ii = id X i , for every i ∈ N ,ϕ ik = ϕ jk ◦ ϕ ij , for every i, j, k ∈ N with i ≤ j ≤ k, where id X i : X i → X i stands for the identity map X i ∋ x x ∈ X i . Moreover, by a target ofthe direct system (cid:0) { ( X i , d i , m i , p i ) } i ∈ N , { ϕ ij } i ≤ j (cid:1) we mean a couple (cid:0) ( Y, d Y , m Y , p Y ) , { ψ i } i ∈ N (cid:1) ,where ( Y, d Y , m Y , p Y ) is a pointed Polish metric measure space, while the maps ψ i : X i → Y are morphisms of pointed Polish metric measure spaces satisfying X i X j Y ϕ ij ψ i ψ j for every i, j ∈ N with i ≤ j . Finally, we say that a target (cid:0) ( X, d X , m X , p X ) , { ϕ i } i ∈ N (cid:1) is the direct limit of (cid:0) { ( X i , d i , m i , p i ) } i ∈ N , { ϕ ij } i ≤ j (cid:1) if it satisfies the following universal property: given any target (cid:0) ( Y, d Y , m Y , p Y ) , { ψ i } i ∈ N (cid:1) , there exists a unique morphism Φ : X → Y ofpointed Polish metric measure spaces such that X i XY ϕ i ψ i Φ for every i ∈ N . Whenever the direct limit exists, it is uniquely determined up to uniqueisomorphism. With an abuse of notation, we will occasionally say that ( X, d X , m X , p X ) is thedirect limit, omitting the reference to the associated morphisms { ϕ i } i ∈ N .A couple (cid:0) { ( X i , d i , m i , p i ) } i ∈ N , { P ij } i ≤ j (cid:1) is said to be an inverse system of pointed Polishmetric measure spaces provided each ( X i , d i , m i , p i ) is a pointed Polish metric measure spaceand the maps P ij : X j → X i are morphisms of pointed Polish metric measure spaces satisfying P ii = id X i , for every i ∈ N ,P ik = P ij ◦ P jk , for every i, j, k ∈ N with i ≤ j ≤ k. Moreover, we say that a given couple (cid:0) ( Y, d Y , m Y , p Y ) , { Q i } i ∈ N (cid:1) projects on the inverse system (cid:0) { ( X i , d i , m i , p i ) } i ∈ N , { P ij } i ≤ j (cid:1) provided ( Y, d Y , m Y , p Y ) is a pointed Polish metric measurespace, while the maps Q i : Y → X i are morphisms of pointed Polish metric measure spacessuch that YX j X iQ j Q i P ij holds for every i, j ∈ N with i ≤ j . The morphisms { Q i } i ∈ N are typically called projec-tions or bonding maps . Finally, a couple (cid:0) ( X, d X , m X , p X ) , { P i } i ∈ N (cid:1) that projects on thedirect system (cid:0) { ( X i , d i , m i , p i ) } i ∈ N , { P ij } i ≤ j (cid:1) is said to be its inverse limit if it satisfies thefollowing universal property: given any couple (cid:0) ( Y, d Y , m Y , p Y ) , { Q i } i ∈ N (cid:1) that projects on (cid:0) { ( X i , d i , m i , p i ) } i ∈ N , { P ij } i ≤ j (cid:1) , there exists a unique morphism Φ : Y → X of pointed Polishmetric measure spaces such that YX X i Φ Q i P i holds for every i ∈ N . Whenever the inverse limit exists, it is uniquely determined up tounique isomorphism.14. Direct limits of pointed metric measure spaces
To begin with, let us study direct limits in the category of pointed Polish metric measurespaces.
Theorem 14.1 (Direct limits of pointed metric measure spaces) . Consider a direct system (cid:16)(cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ N , { ϕ ij } i ≤ j (cid:17) (14.1) of pointed Polish metric measure spaces. Then the direct system in (14.1) admits a directlimit ( X, d X , m X , p X ) if and only if (cid:0) ( X i , d i , m i , p i ) (cid:1) is ω -uniformly boundedly finite. In thiscase, ( X, d X , m X , p X ) = (cid:0) spt( m ω ) , d ω | spt( m ω ) × spt( m ω ) , m ω | B (spt( m ω )) , p ω (cid:1) , where ( X ω , d ω , m ω , p ω ) stands for the ultralimit lim i → ω ( X i , d i , m i , p i ) .Proof. We subdivide the proof into several steps:
Step 1.
First of all, we claim that if (cid:0) ( X i , d i , m i , p i ) (cid:1) is not ω -uniformly boundedly finite, thenthe direct system in (14.1) does not admit any target in the category of pointed Polish metricmeasure spaces, thus in particular it does not have a direct limit in such category. To thisaim, suppose lim i → ω m i (cid:0) B ( p i , R ) (cid:1) = + ∞ for some R >
0. Then we can find a subsequence( i j ) j such that lim j →∞ m i j (cid:0) B ( p i j , R ) (cid:1) = + ∞ . We argue by contradiction: suppose to have atarget (cid:0) ( Y, d Y , m Y , p Y ) , { ψ i } i ∈ N (cid:1) of the direct system in (14.1). Since ψ i | spt( m i ) is 1-Lipschitzand ( ψ i ) ∗ m i ≤ m Y for all i ∈ N , we have that ψ i (cid:0) spt( m i ) ∩ B ( p i , R ) (cid:1) ⊂ B ( p Y , R ) and thus m i j (cid:0) B ( p i j , R ) (cid:1) ≤ ( ψ i j ) ∗ m i j (cid:0) ψ i j (cid:0) spt( m i j ) ∩ B ( p i j , R ) (cid:1)(cid:1) ≤ m Y (cid:0) B ( p Y , R ) (cid:1) , for every j ∈ N . By letting j → ∞ , we conclude that m Y (cid:0) B ( p Y , R ) (cid:1) = + ∞ , which is contradiction with thefact that m Y is boundedly finite. Therefore, no target exists and thus the claim is proven. Step 2.
Hereafter, we shall consider the case where (cid:0) ( X i , d i , m i , p i ) (cid:1) is a ω -uniformly bound-edly finite sequence. Let us define ˆ X i := spt( m i ) ⊂ X i for every i ∈ N . Notice that theinclusion ϕ ij ( ˆ X i ) ⊂ ˆ X j holds for every i, j ∈ N with i ≤ j . Let i ∈ N be fixed. We definethe map ¯ ϕ i : ˆ X i → O ( ¯ X ω ) as ¯ ϕ i ( x i ) := (cid:2) ϕ ik ( x i ) (cid:3) k ≥ i for every x i ∈ ˆ X i , while ϕ i : ˆ X i → X ω isgiven by ϕ i := π ◦ ¯ ϕ i , namely, ϕ i ( x i ) = (cid:2)(cid:2) ϕ ik ( x i ) (cid:3)(cid:3) k ≥ i , for every x i ∈ ˆ X i . We also extend ϕ i to a measurable map defined on the whole X i , by declaring that ϕ i ( x i ) := p ω for every x i ∈ X i \ ˆ X i . Observe that ¯ ϕ i is well-posed, as a consequence of the estimate¯ d ω (cid:0) ¯ ϕ i ( x i ) , [ p k ] k (cid:1) = lim k → ω d k (cid:0) ϕ ik ( x i ) , p k (cid:1) = lim k → ω d k (cid:0) ϕ ik ( x i ) , ϕ ik ( p i ) (cid:1) ≤ d i ( x i , p i ) < + ∞ . Notice also that ϕ i ( p i ) = (cid:2)(cid:2) ϕ ik ( p i ) (cid:3)(cid:3) k ≥ i = [[ p k ]] = p ω . Moreover, we have that ¯ ϕ i (and thusalso ϕ i ) is 1-Lipschitz when restricted to ˆ X i , as shown by the following estimate:¯ d ω (cid:0) ¯ ϕ i ( x i ) , ¯ ϕ i ( y i ) (cid:1) = lim k → ω d k (cid:0) ϕ ik ( x i ) , ϕ ik ( y i ) (cid:1) ≤ d i ( x i , y i ) , for every x i , y i ∈ ˆ X i . Step 3.
Let us now define the closed, separable subset X of X ω as X := closure of [ i ∈ N ϕ i ( ˆ X i ) in X ω . LTRALIMITS OF POINTED METRIC MEASURE SPACES 49
We call d X := d ω | X × X , m X := m ω | B ( X ) , and p X := p ω . Note that ϕ i ( X i ) ⊂ X for all i ∈ N .We aim to prove that each ϕ i : X i → X satisfies ( ϕ i ) ∗ m i ≤ m X . To do so, we first claim thatΠ k → ω ϕ ik ( K ) ⊂ cl O ( ¯ X ω ) (cid:0) ¯ ϕ i ( K ) (cid:1) , for every K ⊂ ˆ X i compact. (14.2)Observe that the set ¯ ϕ i ( K ) is compact (since the map ¯ ϕ i is continuous), but is not necessarilyclosed; this is due to the fact that (cid:0) O ( ¯ X ω ) , ¯ d ω (cid:1) is a pseudometric space and not a metricspace. Fix any ε > C ε the closed ε -neighbourhood of ¯ ϕ i ( K ). Being the set K compact, we can find points z , . . . , z n ∈ K such that K ⊂ S nj =1 B ( z j , ε ). Now fix any[ y k ] ∈ Π k → ω ϕ ik ( K ). Pick some S ∈ ω such that k ≥ i and y k ∈ ϕ ik ( K ) for every k ∈ S .Given any k ∈ S , we can choose x k ∈ K such that y k = ϕ ik ( x k ). Then let us define S j := (cid:8) k ∈ S (cid:12)(cid:12) d i ( x k , z j ) < ε (cid:9) , for every j = 1 , . . . , n. Since S = S nj =1 S j and S ∈ ω , we deduce that S j ∈ ω for some j = 1 , . . . , n . In particular,we have that d i ( x k , z j ) < ε holds for ω -a.e. k ≥ i , thus accordingly¯ d ω (cid:0) [ y k ] k , ¯ ϕ i ( z j ) (cid:1) = lim k → ω d k (cid:0) ϕ ik ( x k ) , ϕ ik ( z j ) (cid:1) ≤ lim k → ω d i ( x k , z j ) ≤ ε. This shows that [ y k ] k ∈ ¯ B (cid:0) ¯ ϕ i ( z j ) , ε (cid:1) ⊂ C ε . Hence, we have proven that Π k → ω ϕ ik ( K ) ⊂ C ε .Since the intersection T ε> C ε coincides with the closure of ¯ ϕ i ( K ) in O ( ¯ X ω ), we obtain (14.2).Let us now pass to the verification of the inequality ( ϕ i ) ∗ m i ≤ m X . Fix any C ⊂ X closed.Since m i is inner regular and ϕ − i ( C ) ∈ B ( X i ), we can find a sequence ( K n ) n of compactsubsets of ϕ − i ( C ) ∩ ˆ X i such that m i (cid:0) ϕ − i ( C ) (cid:1) = lim n →∞ m i ( K n ). Observe that we haveΠ k → ω ϕ ik ( K n ) ⊂ cl O ( ¯ X ω ) (cid:0) ¯ ϕ i ( K n ) (cid:1) ⊂ cl O ( ¯ X ω ) (cid:0) ¯ ϕ i (cid:0) ϕ − i ( C ) ∩ ˆ X i (cid:1)(cid:1) ⊂ π − ( C ), where we used(14.2) and the fact that π − ( C ) is closed thanks to the continuity of π . Then it holds that m X ( C ) = m ω ( C ) = ¯ m ω (cid:0) π − ( C ) (cid:1) ≥ ¯ m ω (cid:0) Π k → ω ϕ ik ( K n ) (cid:1) = lim k → ω m k (cid:0) ϕ ik ( K n ) (cid:1) ≥ lim k → ω ( ϕ k ) ∗ m i (cid:0) ϕ ik ( K n ) (cid:1) = lim k → ω m i (cid:0) ϕ − ik (cid:0) ϕ ik ( K n ) (cid:1)(cid:1) ≥ m i ( K n ) . By letting n → ∞ , we thus deduce that m X ( C ) ≥ m i (cid:0) ϕ − i ( C ) (cid:1) = ( ϕ i ) ∗ m i ( C ). By exploitingthe inner regularity of m X and ( ϕ i ) ∗ m i , we finally conclude that ( ϕ i ) ∗ m i ≤ m X , as desired. Step 4.
Let us now prove that X = spt( m ω ) . (14.3)Given that ϕ i | ˆ X i is 1-Lipschitz (by Step 2 ) and ( ϕ i ) ∗ m i ≤ m ω (by Step 3 ), we can deducethat ϕ i ( ˆ X i ) ⊂ spt( m ω ) for every i ∈ N , thus obtaining that X ⊂ spt( m ω ). In order to provethe converse inclusion, it is enough to show that x / ∈ spt( m ω ) for any given point x ∈ X ω \ X .Fix any r > X ∩ B ( x, r ) = ∅ . Choose any R > B ( x, r ) ⊂ B ( p ω , R/ ϕ ik | ˆ X i is a 1-Lipschitz map and ( ϕ ik ) ∗ m i ≤ m k for all i ≤ k , we have that ϕ ik (cid:0) ˆ X i ∩ B ( p i , R ) (cid:1) ⊂ B ( p k , R ) , for every i, k ∈ N with i ≤ k. (14.4)In particular, if i, k ∈ N satisfy i ≤ k , then it holds that m i (cid:0) B ( p i , R ) (cid:1) (14.4) ≤ m i (cid:0) ϕ − ik (cid:0) B ( p k , R ) (cid:1)(cid:1) = ( ϕ ik ) ∗ m i (cid:0) B ( p k , R ) (cid:1) ≤ m k (cid:0) B ( p k , R ) (cid:1) , which shows that N ∋ i m i (cid:0) B ( p i , R ) (cid:1) is non-decreasing. Thanks to the ω -uniform boundedfiniteness assumption, we have that M := lim i → ω m i (cid:0) B ( p i , R ) (cid:1) < + ∞ . Given any ε >
0, wecan thus find a family S ∈ ω such that M − ε ≤ m i (cid:0) B ( p i , R ) (cid:1) ≤ M, for every i ∈ S. (14.5)For any i ∈ S , pick a compact set K i ⊂ ˆ X i ∩ B ( p i , R ) such that m i (cid:0) B ( p i , R ) \ K i (cid:1) ≤ ε. (14.6)Set i := min( S ). Note that Π k → ω ϕ i k ( K i ) ⊂ cl O ( ¯ X ω ) (cid:0) ¯ ϕ i ( K i ) (cid:1) by (14.2). Calling x = [[ x k ]],we have that Π k → ω B ( x k , r ) ⊂ B (cid:0) [ x k ] , r (cid:1) . Since X ∩ B ( x, r ) = ∅ , we deduce thatΠ k → ω (cid:0) ϕ i k ( K i ) ∩ B ( x k , r ) (cid:1) = (cid:0) Π k → ω ϕ i k ( K i ) (cid:1) ∩ (cid:0) Π k → ω B ( x k , r ) (cid:1) ⊂ cl O ( ¯ X ω ) (cid:0) ¯ ϕ i ( K i ) (cid:1) ∩ B (cid:0) [ x k ] , r (cid:1) ⊂ cl O ( ¯ X ω ) (cid:16) π − (cid:0) ϕ i ( K i ) (cid:1)(cid:17) ∩ π − (cid:0) B ( x, r ) (cid:1) ⊂ π − (cid:16) cl X ω (cid:0) ϕ i ( K i ) (cid:1)(cid:1) ∩ B ( x, r ) (cid:17) ⊂ π − (cid:0) X ∩ B ( x, r ) (cid:1) = ∅ , so that there is a family T ∈ ω such that T ⊂ S and ϕ i k ( K i ) ∩ B ( x k , r ) = ∅ for all k ∈ T .Since Π k → ω B ( x k , r ) ⊂ B (cid:0) [ x k ] , r (cid:1) ⊂ B (cid:0) [ p k ] , R/ (cid:1) ⊂ Π k → ω B ( p k , R ), we can additionallyrequire that B ( x k , r ) ⊂ B ( p k , R ) for every k ∈ T . Therefore, for any k ∈ T it holds that m k (cid:0) B ( x k , r ) (cid:1) ≤ m k (cid:0)(cid:0) ˆ X k ∩ B ( p k , R ) (cid:1) \ ϕ i k ( K i ) (cid:1) (14.4) = m k (cid:0) B ( p k , R ) (cid:1) − m k (cid:0) ϕ i k ( K i ) (cid:1) (14.5) ≤ M − m k (cid:0) ϕ i k ( K i ) (cid:1) ≤ M − ( ϕ i k ) ∗ m i (cid:0) ϕ i k ( K i ) (cid:1) ≤ M − m i ( K i ) (14.6) ≤ M − m i (cid:0) B ( p i , R ) (cid:1) + ε (14.5) ≤ M − ( M − ε ) + ε = 2 ε. Consequently, by using the fact that B (cid:0) [ x k ] , r (cid:1) ⊂ Π k → ω B ( x k , r ), we deduce that m ω (cid:0) B ( x, r ) (cid:1) ≤ ¯ m ω (cid:0) Π k → ω B ( x k , r ) (cid:1) = lim k → ω m k (cid:0) B ( x k , r ) (cid:1) ≤ ε. By the arbitrariness of ε >
0, we conclude that m ω (cid:0) B ( x, r ) (cid:1) = 0, which shows that x doesnot belong to spt( m ω ). Hence, the claimed identity (14.3) is finally proven. Step 5.
Observe that ϕ i = ϕ j ◦ ϕ ij holds for every i, j ∈ N such that i ≤ j . Indeed, we have ϕ i ( x i ) = (cid:2)(cid:2) ϕ ik ( x i ) (cid:3)(cid:3) k ≥ i = (cid:2)(cid:2) ( ϕ jk ◦ ϕ ij )( x i ) (cid:3)(cid:3) k ≥ j = ϕ j (cid:0) ϕ ij ( x i ) (cid:1) , for every x i ∈ ˆ X i . Therefore, (cid:0) ( X, d X , m X , p X ) , { ϕ i } i ∈ N (cid:1) is a target of the direct system in (14.1). In orderto prove that it is actually the direct limit, we have to show that is satisfies the universalproperty. To this aim, fix any target (cid:0) ( Y, d Y , m Y , p Y ) , { ψ i } i ∈ N (cid:1) of the direct system in (14.1).We want to prove that there exists a unique morphism Φ : X → Y of pointed Polish metricmeasure spaces such that ψ i = Φ ◦ ϕ i holds for every i ∈ N . First, notice that for any x ∈ S i ∈ N ϕ i ( ˆ X i ) the choice of Φ( x ) is forced: given i ∈ N and x i ∈ ˆ X i with x = ϕ i ( x i ), we LTRALIMITS OF POINTED METRIC MEASURE SPACES 51 must set Φ( x ) := ψ i ( x i ). We need to check that this definition is well-posed. Namely, we haveto prove that i, j ∈ N , x i ∈ ˆ X i , x j ∈ ˆ X j , ϕ i ( x i ) = ϕ j ( x j ) = ⇒ ψ i ( x i ) = ψ j ( x j ) . (14.7)By using the fact that ψ i ( x i ) = ψ k (cid:0) ϕ ik ( x i ) (cid:1) and ψ j ( x j ) = ψ k (cid:0) ϕ jk ( x j ) (cid:1) for all k ≥ i, j , we get d Y (cid:0) ψ i ( x i ) , ψ j ( x j ) (cid:1) = lim k → ω d Y (cid:0) ψ k (cid:0) ϕ ik ( x i ) (cid:1) , ψ k (cid:0) ϕ jk ( x j ) (cid:1)(cid:1) ≤ lim k → ω d k (cid:0) ϕ ik ( x i ) , ϕ jk ( x j ) (cid:1) = d X (cid:0) ϕ i ( x i ) , ϕ j ( x j ) (cid:1) = 0 , whence (14.7) follows. A similar computation shows that Φ : S i ∈ N ϕ i ( ˆ X i ) → Y is 1-Lipschitz:if x = ϕ i ( x i ) and y = ϕ j ( y j ) for some i, j ∈ N , x i ∈ ˆ X i , and y j ∈ ˆ X j , then we have that d Y (cid:0) Φ( x ) , Φ( y ) (cid:1) = d Y (cid:0) ψ i ( x i ) , ψ j ( y j ) (cid:1) = lim k → ω d Y (cid:0) ψ k (cid:0) ϕ ik ( x i ) (cid:1) , ψ k (cid:0) ϕ jk ( y j ) (cid:1)(cid:1) ≤ lim k → ω d k (cid:0) ϕ ik ( x i ) , ϕ jk ( y j ) (cid:1) = d X (cid:0) ϕ i ( x i ) , ϕ j ( y j ) (cid:1) = d X ( x, y ) . Consequently, it holds that Φ can be uniquely extended to a 1-Lipschitz mapping Φ : X → Y .Moreover, notice that Φ( p X ) = ψ ( p ) = p Y . To prove that Φ is a morphism of pointed Polishmetric measure spaces, it suffices to show that Φ ∗ m X ≤ m Y . To achieve this goal, we needthe following fact: π − (cid:0) Φ − ( B ) (cid:1) ⊂ Π k → ω ψ − k ( B ) , for every B ∈ B ( Y ) . (14.8)Let [ y k ] ∈ π − (cid:0) Φ − ( B ) (cid:1) be given. Choose any i ∈ N and x i ∈ ˆ X i such that [[ y k ]] = ϕ i ( x i ).For every k ≥ i , we know that ψ k (cid:0) ϕ ik ( x i ) (cid:1) = ψ i ( x i ) = Φ (cid:0) [[ y k ]] (cid:1) ∈ B by definition of Φ. Thisimplies that y k = ϕ ik ( x i ) ∈ ψ − k ( B ) holds for ω -a.e. k ≥ i , which is equivalent to saying thatthe element [ y k ] belongs to Π k → ω ψ − k ( B ). Therefore, the claim in (14.8) is proven.Finally, given any Borel set B ⊂ Y , we have thatΦ ∗ m X ( B ) = m ω (cid:0) Φ − ( B ) (cid:1) = ¯ m ω (cid:0) π − (Φ − ( B )) (cid:1) (14.8) ≤ ¯ m ω (cid:0) Π k → ω ψ − k ( B ) (cid:1) = lim k → ω m k (cid:0) ψ − k ( B ) (cid:1) = lim k → ω ( ψ k ) ∗ m k ( B ) ≤ m Y ( B ) . This yields Φ ∗ m X ≤ m Y , thus the universal property is verified. The statement follows. (cid:3) We preferred to formulate Theorem 14.1 in the language of ultralimits, as they constitutethe main topic of this paper, but we can readily provide an alternative (and more explicit)characterisation using the wpmGH convergence.
Corollary 14.2.
A given direct system (14.1) of pointed Polish metric measure spaces admitsa direct limit ( X, d X , m X , p X ) if and only if (cid:0) ( X i , d i , m i , p i ) (cid:1) is uniformly boundedly finite.In this case, ( X, d X , m X , p X ) coincides with the wpmGH limit of (cid:0) ( X i , d i , m i , p i ) (cid:1) .Proof. For a direct system of pointed Polish metric measure spaces, uniform bounded finite-ness and ω -uniform bounded finiteness are equivalent, thanks to the monotonicity of thefunction N ∋ i m i (cid:0) B ( p i , R ) (cid:1) for any R >
Step 4 of Theorem 14.1). Thenthe statement can be proven by combining Theorem 14.1 with the results in Section 11. (cid:3)
Inverse limits of pointed metric measure spaces
We now pass to the study of inverse limits in the category of pointed Polish metric measurespaces.
Theorem 15.1 (Inverse limits of pointed metric measure spaces) . Consider an inverse system (cid:16)(cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ N , { P ij } i ≤ j (cid:17) (15.1) of pointed Polish metric measure spaces. Then (cid:0) ( X i , d i , m i , p i ) (cid:1) is a uniformly boundedly finitesequence, so that the ultralimit ( X ω , d ω , m ω , p ω ) = lim i → ω ( X i , d i , m i , p i ) exists. Let us define X := ˆ X ∩ spt( m ω ) , where ˆ X := n [[ x i ]] ∈ X ω (cid:12)(cid:12)(cid:12) P ij ( x j ) = x i ∈ spt( m i ) for all i ≤ j o , (15.2) while d X := d ω | X × X , m X := m ω | B ( X ) , and p X := p ω . Then ( X, d X , m X ) is a Polish metricmeasure space. Moreover, it holds that the inverse system in (15.1) admits an inverse limitif and only if p X ∈ spt( m X ) . In this case, the inverse limit coincides with ( X, d X , m X , p X ) ,the natural projection maps P i : X → X i being given by P i (cid:0) [[ x k ]] (cid:1) := x i for every [[ x k ]] ∈ X .Proof. We subdivide the proof into several steps:
Step 1.
First of all, let us prove that the sequence (cid:0) ( X i , d i , m i , p i ) (cid:1) is uniformly boundedlyfinite, thus the ultralimit ( X ω , d ω , m ω , p ω ) = lim i → ω ( X i , d i , m i , p i ) exists. Given any i, j ∈ N with i ≤ j , it follows from the fact that P ij | spt( m j ) is 1-Lipschitz that P ij (cid:0) B ( x j , r ) ∩ spt( m j ) (cid:1) ⊂ B (cid:0) P ij ( x j ) , r (cid:1) , for every x j ∈ X j and r > . (15.3)Therefore, since we know that ( P ij ) ∗ m j ≤ m i , we deduce that m j (cid:0) B ( x j , r ) (cid:1) = m j (cid:0) B ( x j , r ) ∩ spt( m j ) (cid:1) (15.3) ≤ m j (cid:0) P − ij (cid:0) B ( P ij ( x j ) , r ) (cid:1)(cid:1) ≤ m i (cid:0) B ( P ij ( x j ) , r ) (cid:1) . (15.4)As a consequence of (15.4), we have that sup i ∈ N m i (cid:0) B ( p i , R ) (cid:1) ≤ m (cid:0) B ( p , R ) (cid:1) < + ∞ holdsfor every R >
0, which proves that (cid:0) ( X i , d i , m i , p i ) (cid:1) is a uniformly boundedly finite sequence. Step 2.
We claim that ˆ X is a closed subset of X ω . (15.5)In order to prove it, we will show that any d ω -Cauchy sequence ( x k ) k ⊂ ˆ X converges to someelement of ˆ X . Say that x k = [[ x ki ]] i , where x ki = P ij ( x kj ) for all i ≤ j . Given any k, k ′ ∈ N ,we have that d i ( x ki , x k ′ i ) = d i (cid:0) P ij ( x ki ) , P ij ( x k ′ i ) (cid:1) ≤ d j ( x kj , x k ′ j ) for all i, j ∈ N with i ≤ j . Thisshows that N ∋ i d i ( x ki , x k ′ i ) is non-decreasing, thus in particular d ω ( x k , x k ′ ) = sup i ∈ N d i ( x ki , x k ′ i ) , for every k, k ′ ∈ N . (15.6)Hence, given any i ∈ N , we know from (15.6) and the fact that ( x k ) k is d ω -Cauchy that ( x ki ) k is d i -Cauchy. Let us denote by x i ∈ spt( m i ) its limit. Define x := [[ x i ]] ∈ X ω . Since P ij iscontinuous, by letting k → ∞ in the identity x ki = P ij ( x kj ) we get x i = P ij ( x j ) for all i ≤ j ,whence it follows that x ∈ ˆ X . It only remains to show that d ω ( x k , x ) → k → ∞ . To thisaim, fix ε >
0. Pick any ¯ k ∈ N such that d ω ( x k , x k ′ ) < ε for all k, k ′ ≥ ¯ k . Given any i ∈ N , we LTRALIMITS OF POINTED METRIC MEASURE SPACES 53 deduce from (15.6) that d i ( x ki , x k ′ i ) < ε for all k, k ′ ≥ ¯ k . Hence, by letting k ′ → ∞ we deducethat d i ( x ki , x i ) ≤ ε for all k ≥ ¯ k , so that d ω ( x k , x ) = lim i → ω d i ( x ki , x i ) ≤ ε for every k ≥ ¯ k .This proves that lim k →∞ d ω ( x k , x ) = 0, thus accordingly the claimed property (15.5) holds.In particular, since spt( m ω ) is complete and separable, we deduce that ( X, d X ) is a Polishmetric space and thus m X is a (well-defined) non-negative Borel measure on X . Step 3.
Now, for any given i ∈ N let us define the projection map P i : X → X i as P i (cid:0) [[ x k ]] (cid:1) := x i , for every [[ x k ]] ∈ X. Suppose that spt( m X ) = ∅ . Observe that P i = P ik ◦ P k for every i, k ∈ N with i ≤ k byconstruction. We now aim to prove that for every i ∈ N it holds that P i | spt( m X ) : spt( m X ) → X i , is 1-Lipschitz , (15.7a)( P i ) ∗ m X ≤ m i . (15.7b)We begin with the verification of (15.7a). Fix [[ x k ]] , [[ y k ]] ∈ spt( m X ). Since x k , y k ∈ spt( m k )and P ik | spt( m k ) is 1-Lipschitz for every k ≥ i , we get d i ( x i , y i ) ≤ d k ( x k , y k ) for every k ≥ i andthus d i ( x i , y i ) ≤ lim k → ω d k ( x k , y k ) = d X (cid:0) [[ x k ]] , [[ y k ]] (cid:1) , which gives (15.7a).In order to prove (15.7b), we need to show that m X (cid:0) P − i ( A ) (cid:1) ≤ m i ( A ) for every givenBorel set A ⊂ X i . Notice that if [ x k ] ∈ π − (cid:0) P − i ( A ) (cid:1) , then P ik ( x k ) = x i ∈ A for ω -a.e. k ≥ i .This shows that π − (cid:0) P − i ( A ) (cid:1) ⊂ Π k → ω P − ik ( A ), so that accordingly m X (cid:0) P − i ( A ) (cid:1) = ¯ m ω (cid:0) π − (cid:0) P − i ( A ) (cid:1)(cid:1) ≤ ¯ m ω (cid:0) Π k → ω P − ik ( A ) (cid:1) = lim k → ω ( P ik ) ∗ m k ( A ) ≤ m i ( A ) . Therefore, (15.7b) is proven. Notice also that if p X ∈ spt( m X ), then P i ( p X ) = p i for all i ∈ N . Step 4.
Now suppose to have a pointed Polish metric measure space (
Y, d Y , m Y , p Y ) andprojection maps Q i : Y → X i for every i ∈ N ; namely, the maps Q i are morphisms of pointedPolish metric measure spaces and Q i = P ij ◦ Q j holds whenever i ≤ j . Without loss ofgenerality, we can assume that spt( m Y ) = Y . Define Φ : Y → ˆ X as Φ( y ) := (cid:2)(cid:2) Q i ( y ) (cid:3)(cid:3) forevery y ∈ Y . By arguing as in Step 1 , we can see that the map Φ : Y → ˆ X is 1-Lipschitzand m Y (cid:0) B ( y, r ) (cid:1) ≤ m i (cid:0) B ( Q i ( y ) , r ) (cid:1) for all i ∈ N , y ∈ Y , and r >
0. In particular, one has m ω (cid:0) ¯ B (Φ( y ) , r ) (cid:1) = inf ℓ ∈ N ¯ m ω (cid:0) Π i → ω B ( Q i ( y ) , r + 1 /ℓ ) (cid:1) = inf ℓ ∈ N lim i → ω m i (cid:0) B ( Q i ( y ) , r + 1 /ℓ ) (cid:1) ≥ m Y (cid:0) B ( y, r ) (cid:1) > , for every r > . This grants that Φ( y ) ∈ spt( m ω ) and thus Φ( Y ) ⊂ X . Observe that Φ is the unique mapfrom Y to X satisfying Q i = P i ◦ Φ for every i ∈ N . Moreover, let us define ¯Φ : Y → O ( ¯ X ω )as ¯Φ( y ) := (cid:2) Q i ( y ) (cid:3) for every y ∈ Y . Notice that Φ = π ◦ ¯Φ. By arguing as in Step 3 of theproof of Theorem 14.1 – specifically, where we proved the inclusion (14.2) – one can showthat Π i → ω Q i ( K ) ⊂ cl O ( ¯ X ω ) (cid:0) ¯Φ( K ) (cid:1) , for every K ⊂ Y compact. (15.8) Consequently, given any closed set C ⊂ X and any compact set K ⊂ Φ − ( C ), it holds that m Y ( K ) ≤ lim i → ω m Y (cid:0) Q − i (cid:0) Q i ( K ) (cid:1)(cid:1) = lim i → ω ( Q i ) ∗ m Y (cid:0) Q i ( K ) (cid:1) ≤ lim i → ω m i (cid:0) Q i ( K ) (cid:1) = ¯ m ω (cid:0) Π i → ω Q i ( K ) (cid:1) (15.8) ≤ ¯ m ω (cid:0) cl O ( ¯ X ω ) (cid:0) ¯Φ( K ) (cid:1)(cid:1) ≤ m ω ( C ) , where in the last inequality we used the fact that π − ( C ) is closed (by continuity of π ) andthus cl O ( ¯ X ω ) (cid:0) ¯Φ( K ) (cid:1) ⊂ cl O ( ¯ X ω ) (cid:0) ¯Φ (cid:0) Φ − ( C ) (cid:1)(cid:1) ⊂ π − ( C ). Thanks to the inner regularity of m Y ,we deduce that m Y (cid:0) Φ − ( C ) (cid:1) ≤ m X ( C ), which in turn yields Φ ∗ m Y ≤ m X as a consequenceof the inner regularity of the measures Φ ∗ m Y and m X . Step 5.
We are now in a position to conclude the proof of the statement. First, we claimthat if p X / ∈ spt( m X ), then no pointed Polish metric measure space projects on the inversesystem in (15.1), thus in particular such inverse system does not admit an inverse limit inthe category of pointed Polish metric measure spaces. We argue by contradiction: supposethere exists a pointed Polish metric measure space ( Y, d Y , m Y , p Y ) that projects on the giveninverse system, via some morphisms Q i : Y → X i . Since p X / ∈ spt( m X ), we can choose aradius r > m X (cid:0) B ( p X , r ) (cid:1) = 0. By using what we proved in Step 4 , we obtain m Y (cid:0) B ( p Y , r ) (cid:1) ≤ m Y (cid:0) Φ − (cid:0) B ( p X , r ) (cid:1)(cid:1) ≤ m X (cid:0) B ( p X , r ) (cid:1) = 0 . This implies that p Y / ∈ spt( m Y ), which leads to a contradiction. Hence, the claim is proven.Conversely, suppose p X ∈ spt( m X ). Then Step 3 grants that (cid:0) ( X, d X , m X , p X ) , { P i } i ∈ N (cid:1) projects on the inverse system in (15.1), while Step 4 shows the validity of the universalproperty. All in all, we have proven that (cid:0) ( X, d X , m X , p X ) , { P i } i ∈ N (cid:1) is the inverse limit of theinverse system in (15.1). Therefore, the statement is finally achieved. (cid:3) Remark 15.2.
Under the assumption of Theorem 15.1, the following implication triviallyholds: p X ∈ spt( m X ) = ⇒ p ω ∈ spt( m ω ) . (15.9)Note that the fact that p ω belongs to the support of m ω can be equivalently characterised as: p ω ∈ spt( m ω ) ⇐⇒ lim i → ω m i (cid:0) B ( p i , r ) (cid:1) > , for every r > . It follows from the inclusions B (cid:0) [ p i ] , r (cid:1) ⊂ Π i → ω B ( p i , r ) ⊂ B (cid:0) [ p i ] , r (cid:1) , which hold for all r .On the other hand, we are not aware of any alternative characterisation of the fact that p X belongs to spt( m X ). Nevertheless, in the ensuing Example 15.3 we will see that the converseimplication of the one in (15.9) might fail. (cid:4) Example 15.3.
Let us denote by ( e j ) j ∈ N the canonical basis of ℓ ∞ , namely, e j := ( δ ij ) i ∈ N .Given any i ∈ N , we define the subset X i ⊂ ℓ ∞ in the following way: X i := ∞ [ k =1 X ki , where we set X ki := (cid:26) e j k (cid:12)(cid:12)(cid:12)(cid:12) j = 1 , . . . , i (cid:27) for every k ∈ N . LTRALIMITS OF POINTED METRIC MEASURE SPACES 55
Moreover, we define p i := e /i , while we set d i ( x, y ) := k x − y k ℓ ∞ for every x, y ∈ X i and m i := ∞ X k =1 k (cid:18) i X j =1 i δ e j /k (cid:19) ∈ P ( X i ) . The bonding maps { P ij } i ≤ j are given as follows: for any i ∈ N , we set P i,i +1 : X i +1 → X i as P i,i +1 (cid:0) e j − /k (cid:1) = P i,i +1 (cid:0) e j /k (cid:1) := e j /k, for every k ∈ N and j = 1 , . . . , i , while we set P ij := P i,i +1 ◦ . . . ◦ P j − ,j : X j → X i for every i, j ∈ N with i < j . Observe that themaps P ij are 1-Lipschitz and satisfy ( P ij ) ∗ m j = m i . Hence, the sequence (cid:8) ( X i , d i , m i , p i ) (cid:9) i ∈ N together with the maps { P ij } i ≤ j form an inverse system in the category of pointed Polishmetric measure spaces. Let ( X, d X , m X , p X ) be as in Theorem 15.1. Then we claim thatspt( m ω ) = { p ω } , spt( m X ) = ∅ . (15.10)Let us first show that p ω ∈ spt( m ω ). Given any r >
0, choose ¯ k ∈ N such that 2 / ¯ k < r/ i ≥ ¯ k we have m i (cid:0) B ( p i , r/ (cid:1) ≥ P k ≥ ¯ k − k = 2 − ¯ k +1 , so that m ω (cid:0) B ( p ω , r ) (cid:1) ≥ lim i → ω m i (cid:0) B ( p i , r/ (cid:1) ≥ ¯ k − > . By the arbitrariness of r >
0, we deduce that p ω ∈ spt( m ω ). Now fix any x = [[ x i ]] ∈ X ω \{ p ω } .We aim to show that p ω / ∈ spt( m ω ). Since lim i → ω d i ( x i , p i ) = d ω ( x, p ω ) >
0, we can find ¯ k ∈ N with x i ∈ S k ≤ ¯ k X ki for ω -a.e. i . Then there exist k ∈ { , . . . , ¯ k } and S ∈ ω such that x i ∈ X k i for every i ∈ S . Given that B (cid:0) x i , /k ( k + 1) (cid:1) = { x i } and m i (cid:0) { x i } (cid:1) = 2 − k − i for every i ∈ S ,we conclude that m ω (cid:0) B (cid:0) x, /k ( k + 2) (cid:1)(cid:1) ≤ lim i → ω m i (cid:0) B (cid:0) x i , /k ( k + 1) (cid:1)(cid:1) = 0, thus provingthat x / ∈ spt( m ω ). All in all, the first part of (15.10) is proven. To prove the second one, it issufficient to show that m ω (cid:0) { p ω } (cid:1) = 0, since this implies that m X = 0 and thus spt( m X ) = ∅ .To this aim, fix any r >
0. Given that m i (cid:0) B ( p i , r ) (cid:1) ≤ − /r for all i ∈ N with 1 /i < r ,we deduce that m ω (cid:0) { p ω } (cid:1) ≤ lim i → ω m i (cid:0) B ( p i , r ) (cid:1) ≤ − /r . By letting r ց
0, we can finallyconclude that m ω (cid:0) { p ω } (cid:1) = 0, as claimed above. (cid:4) PPENDIX A
Prokhorov theorem for ultralimits
In this appendix, we obtain a variant for ultralimits of the celebrated Prokhorov theorem;see Theorem A.5. Even though we only needed one of the two implications (in the proof ofTheorem 11.1), we prove the full result, since we believe it might be of independent interest.
Lemma A.1.
Let ( X, d ) be a complete metric space. Let ( x i ) i ∈ N ⊂ X be an asymptoticallyCauchy sequence, meaning that for every ε > there exists S ∈ ω such that d ( x i , x j ) ≤ ε holds for every i, j ∈ S . Then the ultralimit x := lim i → ω x i ∈ X exists.Proof. Fix any sequence ε k ց
0. Given any k ∈ N , we can find S ′ k ∈ ω such that d ( x i , x j ) ≤ ε k for every i, j ∈ S ′ k . Define S := S ′ and S k := S ′ k ∩ S k − ∩ (cid:8) i ∈ N : i > min( S k − ) (cid:9) ∈ ω, for every k ≥ . Then the sequence ( i k ) k ∈ N , given by i k := min( S k ), is strictly increasing. Since d ( x i ℓ , x i k ) ≤ ε k for all ℓ ≥ k , we deduce that ( x i k ) k ∈ N is a Cauchy sequence, thus it admits a limit x ∈ X . Weclaim that x = lim i → ω x i . In order to prove it, let ε > k ∈ N such that ε k ≤ ε and d ( x i k , x ) ≤ ε . Since d ( x i , x i k ) ≤ ε k ≤ ε for all i ∈ S k , we have that d ( x i , x ) ≤ ε for ω -a.e. i . By arbitrariness of ε , we conclude that x = lim i → ω x i , as required. (cid:3) Given a Polish metric space (
X, d ) and a constant λ >
0, we define the family M λ ( X ) as M λ ( X ) := (cid:8) µ ≥ X (cid:12)(cid:12) µ ( X ) ≤ λ (cid:9) . By weak topology in duality with C b ( X ) we mean the coarsest topology on M λ ( X ) suchthat the function M λ ( X ) ∋ µ → ´ f d µ ∈ R is continuous for every f ∈ C b ( X ). The weaktopology is metrised by the following distance: δ ( µ, ν ) := X n ∈ N n (cid:12)(cid:12)(cid:12)(cid:12) ˆ f n d( µ − ν ) (cid:12)(cid:12)(cid:12)(cid:12) , for every µ, ν ∈ M λ ( X ) , where ( f n ) n ∈ N is a suitably chosen sequence of bounded, Lipschitz functions from X to R satisfying k f n k C b ( X ) = 1 for all n ∈ N . For instance, one can take as ( f n ) n the countablefamily (cid:8) g/ k g k C b ( X ) (cid:12)(cid:12) g ∈ F , g = 0 (cid:9) , where we define F := (cid:8) ± max { g , . . . , g k } (cid:12)(cid:12) k ∈ N , g , . . . , g k ∈ ˜ F (cid:9) , while (for some given countable, dense subset D of X ) we define˜ F := n max (cid:8) α − β d ( · , y ) , γ (cid:9) (cid:12)(cid:12)(cid:12) α, β, γ ∈ Q , y ∈ D o . Observe that, given any f ∈ C b ( X ), µ ∈ M λ ( X ), and ε >
0, there exist functions g, g ′ ∈ F such that g ≤ f ≤ g ′ and ´ g ′ d µ − ε ≤ ´ f d µ ≤ ´ g d µ + ε . Remark A.2.
Given any Polish metric space (
X, d ), it holds that (cid:0) P ( X ) , δ (cid:1) is a Polishmetric space, where P ( X ) stands for the space of all Borel probability measures on ( X, d ).See, for instance, [2, Theorem 15.15]. (cid:4)
Proposition A.3.
Let ( X, d ) be a Polish metric space. Let µ, µ i ∈ M λ ( X ) for some λ > .Then it holds µ = lim i → ω µ i with respect to the weak convergence in M λ ( X ) if and only if ˆ f d µ = lim i → ω ˆ f d µ i , for every f ∈ C b ( X ) . (A.1) Proof.
Suppose µ = lim i → ω µ i with respect to the weak convergence. Let f ∈ C b ( X ) be fixed.Given ε >
0, we can find g, g ′ ∈ F such that g ≤ f ≤ g ′ and ´ g ′ d µ − ε ≤ ´ f d µ ≤ ´ g d µ + ε .Choose n, m ∈ N with f n = g/ k g k C b ( X ) and f m = g ′ / k g ′ k C b ( X ) . Then for any i ∈ N it holds ˆ f d µ i − ˆ f d µ ≥ k g k C b ( X ) (cid:18) ˆ f n d( µ i − µ ) (cid:19) − ε ≥ − n k g k C b ( X ) δ ( µ i , µ ) − ε, ˆ f d µ i − ˆ f d µ ≤ k g ′ k C b ( X ) (cid:18) ˆ f m d( µ i − µ ) (cid:19) + ε ≤ m k g ′ k C b ( X ) δ ( µ i , µ ) + ε. All in all, for any i ∈ N we have that (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d µ i − ˆ f d µ (cid:12)(cid:12)(cid:12)(cid:12) ≤ c δ ( µ i , µ ) + ε, where c := max (cid:8) n k g k C b ( X ) , m k g ′ k C b ( X ) (cid:9) . Therefore, we conclude that lim i → ω (cid:12)(cid:12) ´ f d µ i − ´ f d µ (cid:12)(cid:12) ≤ c lim i → ω δ ( µ i , µ ) + ε = ε , whence thedesired property (A.1) follows thanks to the arbitrariness of ε .Conversely, suppose (A.1) is verified. Let ε > N ∈ N such that 1 / N ≤ ε/ S := (cid:26) i ∈ N (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ˆ f n d( µ i − µ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε n = 1 , . . . , N (cid:27) ∈ ω. Therefore, we finally conclude that δ ( µ i , µ ) ≤ N X n =1 n (cid:12)(cid:12)(cid:12)(cid:12) ˆ f n d( µ i − µ ) (cid:12)(cid:12)(cid:12)(cid:12) + X n>N n ≤ ε N ≤ ε, for every i ∈ S, so that δ ( µ i , µ ) ≤ ε for ω -a.e. i , thus µ = lim i → ω µ i with respect to the weak convergence. (cid:3) Alternatively, we could have proven Proposition A.3 by just observing that for any givenmeasure µ ∈ M λ ( X ) the collection (cid:8) U f,ε : f ∈ C b ( X ) , ε > (cid:9) , which is given by U f,ε := (cid:26) ν ∈ M λ ( X ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d µ − ˆ f d ν (cid:12)(cid:12)(cid:12)(cid:12) < ε (cid:27) , forms a neighbourhood basis for µ with respect to the weak topology. Corollary A.4.
Let ( X, d ) be a Polish metric space. Let λ > be given. Let µ, µ i ∈ M λ ( X ) satisfy µ = lim i → ω µ i with respect to the weak convergence in M λ ( X ) . Then it holds that µ ( U ) ≤ lim i → ω µ i ( U ) , for every open set U ⊂ X, (A.2a) µ ( C ) ≥ lim i → ω µ i ( C ) , for every closed set C ⊂ X. (A.2b) In particular, it holds that µ ( X ) = lim i → ω µ i ( X ) .Proof. To prove (A.2a), pick a sequence ( f n ) n ∈ N ⊂ C b ( X ) with f n ր χ U everywhere on X .Thanks to Proposition A.3, we see that ˆ f n d µ = lim i → ω ˆ f n d µ i ≤ lim i → ω µ i ( U ) , for every n ∈ N , whence by letting n → ∞ and using the monotone convergence theorem we can conclude that µ ( U ) = lim n →∞ ˆ f n d µ ≤ lim i → ω µ i ( U ) . This shows the validity of (A.2a). To prove (A.2b), we can argue in a similar way. Pick asequence ( g n ) n ∈ N ⊂ C b ( X ) such that g n ≤ n ∈ N and g n ց χ C everywhere on X . Byusing Proposition A.3 we get ´ g n d µ = lim i → ω ´ g n d µ i ≥ lim i → ω µ i ( C ) for all n ∈ N , so bydominated convergence theorem we conclude that µ ( C ) = lim n →∞ ´ g n d µ ≥ lim i → ω µ i ( C ).This shows the validity of (A.2b). The final statement immediately follows. (cid:3) A subset T of a metric space ( X, d ) is said to be ε -totally bounded (for some ε >
0) providedit is contained in the union of finitely many balls of radius ε . Observe that a subset of X istotally bounded if and only if it is ε -totally bounded for every ε > Theorem A.5 (Prokhorov theorem for ultralimits) . Let ( X, d ) be a given Polish metric space.Let ( µ i ) i ∈ N be a sequence of Borel measures on ( X, d ) such that λ := lim i → ω µ i ( X ) < + ∞ .Then the following conditions are equivalent: i) Given any ε > , there exists an ε -totally bounded, Borel subset T of X such that lim i → ω µ i ( X \ T ) ≤ ε. ii) The ultralimit µ := lim i → ω µ i exists (with respect to the weak convergence in dualitywith C b ( X ) ).Proof. The proof goes along the lines of the proof of Prokhorov theorem, see for example [6].ii) = ⇒ i). Suppose ii) holds and fix ε >
0. Choose any dense sequence ( x j ) j ∈ N in X . Thenwe claim that lim i → ω µ i (cid:18) N [ j =1 B ( x j , ε ) (cid:19) ≥ λ − ε, for some N ∈ N . (A.3)We argue by contradiction: suppose that lim i → ω µ i (cid:0) S Nj =1 B ( x j , ε ) (cid:1) < λ − ε for every N ∈ N .Hence, Corollary A.4 grants that µ (cid:0) S Nj =1 B ( x j , ε ) (cid:1) ≤ lim i → ω µ i (cid:0) S Nj =1 B ( x j , ε ) (cid:1) < λ − ε ,whence by letting N → ∞ we get µ ( X ) ≤ λ − ε ; we are using the fact that X = S j ∈ N B ( x j , ε )by density of the sequence ( x j ) j ∈ N . On the other hand, we also know from Corollary A.4 that LTRALIMITS OF POINTED METRIC MEASURE SPACES 59 µ ( X ) = lim i → ω µ i ( X ) = λ , thus leading to a contradiction. This shows the validity of (A.3).Therefore, the Borel set T := S Nj =1 B ( x j , ε ) (where N ∈ N is given as in (A.3)) is ε -totallybounded and satisfies lim i → ω µ i ( X \ T ) = lim i → ω µ i ( X ) − lim i → ω µ i ( T ) ≤ ε , thus yielding i).i) = ⇒ ii). Suppose i) holds. If λ = 0, then we trivially have lim i → ω µ i = 0 with respect to theweak convergence, so that ii) is verified. Hence, we can assume that λ >
0. We claim that(˜ µ i ) i ∈ N is asymptotically Cauchy in (cid:0) P ( X ) , δ (cid:1) , where we set ˜ µ i := µ i µ i ( X ) . (A.4)In order to prove it, fix any ε ∈ (0 , λ ). Pick some N ∈ N for which 1 / N ≤ ε . Define η := ε (cid:8) Lip( f ) , . . . , Lip( f N ) (cid:9) . By using i), we find an η -totally bounded set T such that lim i → ω µ i ( X \ T ) /µ i ( X ) < η ≤ ε .Write T as a disjoint union F ∪ . . . ∪ F k of Borel sets F , . . . , F k = ∅ having diameter smallerthan 2 η . Pick y j ∈ F j for any j = 1 , . . . , k . Calling S ′ := (cid:8) i ∈ N : µ i ( X \ T ) /µ i ( X ) ≤ ε (cid:9) ∈ ω ,we define the Borel measures ( ν i ) i ∈ S ′ on X as ν i := P kj =1 µ i ( F j ) δ y j µ i ( X ) ∈ M ( X ) , for every i ∈ S ′ . If we set λ j := lim i → ω µ i ( F j ) for every j = 1 , . . . , k , then it holds that S := (cid:26) i ∈ S ′ (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) µ i ( F j ) µ i ( X ) − λ j λ (cid:12)(cid:12)(cid:12)(cid:12) ≤ εk for every j = 1 , . . . , k (cid:27) ∈ ω. Let us now define ν := P kj =1 λ j δ y j λ ∈ M ( X ) . Given any n = 1 , . . . , N and i ∈ S , we may estimate (cid:12)(cid:12)(cid:12)(cid:12) ˆ f n d(˜ µ i − ν i ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ i ( X ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ X \ T f n d µ i (cid:12)(cid:12)(cid:12)(cid:12) + 1 µ i ( X ) k X j =1 (cid:12)(cid:12)(cid:12)(cid:12) ˆ F j f n d µ i − µ i ( F j ) f n ( y j ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ i ( X \ T ) µ i ( X ) + 1 µ i ( X ) k X j =1 ˆ F j (cid:12)(cid:12) f n ( x ) − f n ( y j ) (cid:12)(cid:12) d µ i ( x ) ≤ ε + Lip( f n ) µ i ( X ) k X j =1 ˆ F j d ( x, y j ) d µ i ( x ) ≤ ε + 2 η Lip( f n ) µ i ( X ) k X j =1 µ i ( F j ) ≤ ε + ε µ i ( T ) µ i ( X ) ≤ ε. Moreover, it holds that (cid:12)(cid:12)(cid:12)(cid:12) ˆ f n d( ν i − ν ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) k X j =1 f n ( y j ) (cid:18) µ i ( F j ) µ i ( X ) − λ j λ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k X j =1 (cid:12)(cid:12) f n ( y j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ i ( F j ) µ i ( X ) − λ j λ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε. All in all, we obtain (cid:12)(cid:12) ´ f n d(˜ µ i − ν ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) ´ f n d(˜ µ i − ν i ) (cid:12)(cid:12) + (cid:12)(cid:12) ´ f n d( ν i − ν ) (cid:12)(cid:12) ≤ ε for every i ∈ S and n = 1 , . . . , N , thus accordingly we conclude that δ (˜ µ i , ˜ µ ℓ ) = N X n =1 n (cid:12)(cid:12)(cid:12)(cid:12) ˆ f n d(˜ µ i − ˜ µ ℓ ) (cid:12)(cid:12)(cid:12)(cid:12) + X n>N n (cid:12)(cid:12)(cid:12)(cid:12) ˆ f n d(˜ µ i − ˜ µ ℓ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ N X n =1 n (cid:12)(cid:12)(cid:12)(cid:12) ˆ f n d(˜ µ i − ν ) (cid:12)(cid:12)(cid:12)(cid:12) + N X n =1 n (cid:12)(cid:12)(cid:12)(cid:12) ˆ f n d( ν − ˜ µ ℓ ) (cid:12)(cid:12)(cid:12)(cid:12) + X n>N n ≤ ε + 12 N ≤ ε, for every i, ℓ ∈ S. This shows the claim (A.4). Therefore, by using Lemma A.1 and Remark A.2 we get existenceof a measure ˜ µ ∈ P ( X ) such that ˜ µ = lim i → ω ˜ µ i with respect to the weak convergence. Letus define µ := λ ˜ µ ∈ M λ ( X ). Then it can be readily checked that µ = lim i → ω µ i with respectto the weak convergence, whence ii) follows. The proof of the statement is achieved. (cid:3) Proposition A.6 (Sharpness of Prokhorov Theorem A.5) . Let ω be a non-principal ultrafilteron N satisfying the property that is described in Lemma 5.5. Then there exists a Polish metricspace ( X, d ) and a sequence ( µ i ) i ∈ N ⊂ P ( X ) of measures such that µ = lim i → ω µ i (with respectto the weak convergence) for some µ ∈ P ( X ) , but lim i → ω µ i ( K ) = 0 , for every compact set K ⊂ X. (A.5) In particular, this shows that Theorem A.5 might fail if in its item i) we replace “ ε -totallybounded” with “compact”.Proof. Let us define the subset X of ℓ ∞ as X := { } ∪ (cid:26) e j n (cid:12)(cid:12)(cid:12)(cid:12) j, n ∈ N (cid:27) , where e j := ( δ jk ) k ∈ N stands for the j -th element of the canonical basis of ℓ ∞ . Calling d thedistance on X induced by the ℓ ∞ -norm, we have that ( X, d ) is a Polish metric space. Fix apartition { N n } n ∈ N of N into infinite sets as in Lemma 5.5. Say that each set N n is written as { m nk } k ∈ N , where m n < m n < m n < . . . . Let us then define µ m nk ∈ P ( X ) as µ m nk := 1 k k X j =1 δ e j /n , for every n, k ∈ N . First of all, we claim that the resulting sequence ( µ i ) i ∈ N of measures satisfies δ = lim i → ω µ i , with respect to the weak convergence in duality with C b ( X ) . (A.6)In order to prove it, let f ∈ C b ( X ) and ε > r > (cid:12)(cid:12) f ( x ) − f (0) (cid:12)(cid:12) < ε for every x ∈ B (0 , r ). Pick any ¯ n ∈ N such that 1 / ¯ n ≤ r , so that wehave e j /n ∈ B (0 , r ) for all n > ¯ n and j ∈ N . Consequently, for any n > ¯ n and k ∈ N it holds (cid:12)(cid:12)(cid:12)(cid:12) ˆ f d µ m nk − ˆ f d δ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) k k X j =1 f ( e j /n ) − f (0) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k k X j =1 (cid:12)(cid:12) f ( e j /n ) − f (0) (cid:12)(cid:12) ≤ ε. LTRALIMITS OF POINTED METRIC MEASURE SPACES 61
In other words, one has (cid:12)(cid:12) ´ f d µ i − ´ f d δ (cid:12)(cid:12) ≤ ε for every i ∈ N \ N ¯ n . Given that N \ N ¯ n ∈ ω (recall Lemma 5.5), thanks to the arbitrariness of ε > K ⊂ X be a given compact set. It can bereadily checked that necessarily lim j →∞ diam( K ∩ R e j ) = 0. We argue by contradiction:suppose that λ := lim i → ω µ i ( K ) >
0. Then there exists S ∈ ω such that µ i ( K ) ≥ λ/ i ∈ S . By Lemma 5.5 we can find n ∈ N for which S ∩ N n is infinite (otherwise, N m \ ( N \ S )would be finite for all m ∈ N and thus N \ S ∈ ω , contradicting the fact that S ∈ ω ). Choosea subsequence ( k ℓ ) ℓ ∈ N such that m nk ℓ ∈ S holds for every ℓ ∈ N . Now let us fix some ¯ ℓ ∈ N satisfying K ∩ R e k ℓ ⊂ B (0 , /n ) for all ℓ ≥ ¯ ℓ . Therefore, one has that µ m nkℓ ( K ) ≤ k ¯ ℓ /k ℓ forevery ℓ ≥ ¯ ℓ . Since lim ℓ →∞ k ¯ ℓ /k ℓ = 0, we see that inf i ∈ S µ i ( K ) ≤ inf ℓ ∈ N µ m nkℓ ( K ) = 0, whichleads to a contradiction with the fact that inf i ∈ S µ i ( K ) ≥ λ/
2. Hence, (A.5) is proven. (cid:3)
PPENDIX B
Tangents to pointwise doubling metric measure spaces
With the notion of wpmGH-convergence at our disposal, we can now introduce a naturaldefinition of tangent cone to a Polish metric measure space at a fixed point of its support.
Definition B.1 (wpmGH-tangent cone) . Let ( X, d, m , p ) be a pointed Polish metric measurespace with p ∈ spt ( m ) . Then we define Tan(
X, d, m , p ) as the family of all those pointedPolish metric measure spaces ( Y, d Y , m Y , q ) satisfying q ∈ spt ( m Y ) and ( X, d/r i , m pr i , p ) −→ ( Y, d Y , m Y , q ) , in the wpmGH-sense,for some sequence r i ց , where the Borel measure m pr i ≥ over ( X, d/r i ) is given by m pr i := mm (cid:0) B d ( p, r i ) (cid:1) . Note that, trivially, the identity B d/r ( x, s ) = B d ( x, rs ) holds for every x ∈ X and r, s > Definition B.2 (Pointwise doubling) . Let ( X, d, m ) be a Polish metric measure space. Thenwe say that ( X, d, m ) is pointwise doubling provided it holds that lim r ց m (cid:0) B ( x, r ) (cid:1) m (cid:0) B ( x, r ) (cid:1) < + ∞ , for m -a.e. x ∈ X. By suitably adapting the proof of [25, Theorem 1.6], one can see that every pointwisedoubling space (
X, d, m ) is a Vitali space (in the sense of [25, Remark 1.13]), so that Lebesgue’sdifferentiation theorem holds. In particular, given any Borel set E ⊂ X , we have that m -a.e.point x ∈ E is of density 1 for E , meaning that lim r ց m (cid:0) E ∩ B ( x, r ) (cid:1) / m (cid:0) B ( x, r ) (cid:1) = 1. Thefollowing statement about pointwise doubling spaces is proven in [5, Lemma 8.3]. Lemma B.3.
Let ( X, d, m ) be a pointwise doubling Polish metric measure space. Then thereexists a family ( E n ) n ∈ N of pairwise disjoint Borel subsets of X such that m (cid:0) X \ S n ∈ N E n (cid:1) = 0 and (cid:0) E n , d | E n × E n (cid:1) is metrically doubling for every n ∈ N . Remark B.4.
Suppose lim r ց m ( B ( p, r )) m ( B ( p,r )) < + ∞ . Then for any sequence r i ց i → ω m pr i (cid:0) B d/r i ( p, R ) (cid:1) ≤ lim i →∞ m pr i (cid:0) B d ( p, Rr i ) (cid:1) = lim i →∞ m (cid:0) B d ( p, Rr i ) (cid:1) m (cid:0) B d ( p, r i ) (cid:1) < + ∞ , for any R > . Therefore, the sequence (cid:0) ( X, d/r i , m pr i , p ) (cid:1) is ω -uniformly boundedly finite. (cid:4) Theorem B.5 (Existence of wpmGH tangents to pointwise doubling spaces) . Let ( X, d, m ) be a pointwise doubling Polish metric measure space. Let r i ց be given. Then for m -a.e.point p ∈ X the sequence (cid:0) ( X, d/r i , m pr i , p ) (cid:1) is asymptotically boundedly m ω -totally bounded.In particular, it holds that Tan(
X, d, m , p ) = ∅ , for m -a.e. p ∈ X. (B.1) Proof.
Let ( E n ) n ∈ N be chosen as in Lemma B.3. Fix a point p ∈ X that (belongs to and) isof density 1 for E n , for some n ∈ N ; as observed after Definition B.2, m -almost every pointof X has this property. Now let R, r, ε > (cid:0) E n , d | E n × E n (cid:1) is metricallydoubling by Lemma B.3, we can find a constant M ∈ N such that the following property issatisfied: given any s ∈ (0 , E n ) of center p and radius Rs can be covered by M balls of radius rs . Hence, for any i ∈ N there exist x i , . . . , x iM ∈ E n ∩ B d ( p, Rr i ) such that E n ∩ ¯ B d/r i ( p, R ) = E n ∩ ¯ B d ( p, Rr i ) ⊂ M [ j =1 B d ( x ij , rr i ) = M [ j =1 B d/r i ( x ij , r ) . (B.2)By using the fact that p is of density 1 for E n , we thus conclude thatlim i → ω m pr i (cid:0) ¯ B d/r i ( p, R ) \ S Mj =1 B d/r i ( x ij , r ) (cid:1) ≤ lim i →∞ m pr i (cid:0) ¯ B d/r i ( p, R ) \ S Mj =1 B d/r i ( x ij , r ) (cid:1) (B.2) ≤ lim i →∞ m pr i (cid:0) ¯ B d/r i ( p, R ) \ E n (cid:1) = lim i →∞ m (cid:0) ¯ B d ( p, Rr i ) \ E n (cid:1) m (cid:0) B d ( p, Rr i ) (cid:1) m (cid:0) B d ( p, Rr i ) (cid:1) m (cid:0) B d ( p, r i ) (cid:1) = 0 . This shows that (cid:0) ( X, d/r i , m pr i , p ) (cid:1) is asymptotically boundedly m ω -totally bounded. The laststatement (B.1) now immediately follows from the first one together with Theorem 11.4. (cid:3) References [1]
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Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
Email address : [email protected] (Timo Schultz) Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 100131, 33501 Biele-feld, Germany.
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