aa r X i v : . [ m a t h . M G ] A ug CONVERGENCE OF METRIC TRANSFORMED SPACES
DAISUKE KAZUKAWA
Abstract.
We consider the metric transformation of metric measure spaces/pyramids. Weclarify the conditions to obtain the convergence of the sequence of transformed spaces fromthat of the original sequence, and, conversely, to obtain the convergence of the original se-quence from that of the transformed sequence, respectively. As an application, we prove thatspheres and projective spaces with standard Riemannian distance converge to a Gaussianspace and the Hopf quotient of a Gaussian space, respectively, as the dimension diverges toinfinity.
Contents
1. Introduction 12. Preliminaries 53. Metric preserving functions 94. Weak convergence of metric transformed pyramids 155. Box-convergence/concentration of metric transformed spaces 296. Application: spheres and projective spaces 31References 341.
Introduction
The geometry and analysis on metric measure spaces have actively been studied. Met-ric measure spaces typically appear as limit spaces of Riemannian manifolds in the con-vergence/collapsing theory of Riemannian manifolds. The study of convergence of metricmeasure spaces is one of central topics in geometric analysis on metric measure spaces.Gromov [3, Chapter 3
12 + ] has developed a new convergence theory of metric measure spacesbased on the concentration of measure phenomenon studied by L´evy and V. Milman [8, 9](see also [7]) which is roughly stated as that any 1-Lipschitz function on high-dimensionalspaces is close to a constant. Gromov introduced two fundamental concepts of distancefunctions, the observable distance function d conc and the box distance function (cid:3) , on theset, say X , of isomorphism classes of metric measure spaces. The box distance function isnearly a metrization of measured Gromov-Hausdorff convergence (precisely the isomorphism Date : August 10, 2020.2010
Mathematics Subject Classification.
Primary 53C23.
Key words and phrases. metric measure space, pyramid, concentration topology, weak topology, box topol-ogy, metric preserving function.The author is supported by JSPS KAKENHI Grant Number 20J00147.
ONVERGENCE OF METRIC TRANSFORMED SPACES 2 classes are little different), while the observable distance function induces a very charac-teristic topology, called the concentration topology, which admits the convergence of manysequences whose dimensions are unbounded. The concentration topology is weaker than thebox topology and in particular, a measured Gromov-Hausdorff convergence becomes a con-vergence in the concentration topology. He also introduced a natural compactification, sayΠ, of X with respect to the concentration topology, where the topology on Π is called theweak topology. An element P of Π is called a pyramid and is expressed as a subset of X , sothat Π is a subset of the power set of X . Under this compactification, we often identify ametric measure space X with a pyramid, say P X , associated with X . We refer to Section 2for the precise definitions.The study of the concentration and the weak topologies has been growing in recent years(see [2, 4–6, 10–15]). In particular, we have obtained in [6, 14, 15] some nontrivial examplesof weak convergent sequences, for example, spheres with the restriction of Euclidean norm,solid ellipses, (projective) Stiefel manifolds with the Frobenius norm, whose dimensions areunbounded. However, in all these examples, the distance function comes from the Euclideandistance. Our final goal in this paper is to give the first nontrivial example of weak convergentsequences of non-Euclidean Riemannian manifolds. For this purpose, we will investigate theconvergence of metric transformed spaces.Let F : [0 , + ∞ ) → [0 , + ∞ ) be a continuous function satisfying the following condition: forany metric space ( X, d X ), the function F ◦ d X is a metric on X . Such a function F is calleda metric preserving function. The concept of metric preserving functions was discoveredin 1930s and the study of these functions has been deepened. For example, the function s s/ (1 + s ) is a well-known metric preserving function. More generally, if a function F : [0 , + ∞ ) → [0 , + ∞ ) with F − (0) = { } is concave, then F is metric-preserving. However,it is also known that metric preserving functions are not necessarily nondecreasing and theclass of all metric preserving functions is more complicated. We describe some properties ofmetric preserving functions in Section 3. Definition 1.1.
Let F : [0 , + ∞ ) → [0 , + ∞ ) be a continuous metric preserving function.Given a metric measure space X = ( X, d X , m X ), we define a metric measure space F ( X ) := ( X, F ◦ d X , m X ) . We call F ( X ) the metric transformed space of X by F . In addition, for a pyramid P ( ⊂ X ),we define F ( P ) := [ X ∈P P F ( X ) (cid:3) , where Y (cid:3) means the (cid:3) -closure of a family Y of metric measure spaces. If F is nondecreasing, F ( P ) is a pyramid and is called the metric transformed pyramid of a pyramid P by F .We refer to Proposition 4.1 for the proof that F ( P ) is a pyramid if F is nondecreasing. Inthis proposition, we also show that F ( P X ) = P F ( X ) holds for any metric measure space X if F is nondecreasing. However, if not, F ( P ) may not be a pyramid and there exists a metricmeasure space X such that F ( P X ) = P F ( X ).The following theorems are the main results of this paper. ONVERGENCE OF METRIC TRANSFORMED SPACES 3
Theorem 1.2.
Let F n , F : [0 , + ∞ ) → [0 , + ∞ ) , n = 1 , , . . . , be continuous metric preservingfunctions. Assume that F is nondecreasing. Then the following (1) and (2) are equivalent toeach other. (1) For any sequence { X n } n ∈ N of metric measure spaces and for any pyramid P , if X n converges weakly to P , then F n ( X n ) converges weakly to F ( P ) as n → ∞ . (2) The following three conditions hold. (I) F n converges pointwise to F as n → ∞ . (II) For any s ∈ [0 , + ∞ ) , lim n →∞ ( F n ( s ) − inf s ≤ s ′ F n ( s ′ )) = 0 . (III) lim sup n →∞ sup F n ≤ sup F. Theorem 1.3.
Let F n , F : [0 , + ∞ ) → [0 , + ∞ ) , n = 1 , , . . . , be continuous metric preservingfunctions. Assume that F is nondecreasing. Then the following (1) and (2) are equivalent toeach other. (1) For any sequence { X n } n ∈ N of metric measure spaces and for any pyramid P , if F n ( X n ) converges weakly to F ( P ) , then X n converges weakly to P as n → ∞ . (2) The following three conditions hold. (I) F n converges pointwise to F as n → ∞ . (II) For any s ∈ [0 , + ∞ ) , lim n →∞ ( F n ( s ) − inf s ≤ s ′ F n ( s ′ )) = 0 . (IV) F is increasing. If all F n are assumed to be nondecreasing in advance, then we also obtain the followingversion. We remark that the condition (II) is always true if all F n are nondecreasing. Corollary 1.4.
Let F n , F : [0 , + ∞ ) → [0 , + ∞ ) , n = 1 , , . . . , be continuous nondecreasingmetric preserving functions. Then the following (A) and (B) hold. (A) The following (A1) and (A2) are equivalent to each other. (A1)
For any sequence {P n } n ∈ N of pyramids and for any pyramid P , if P n convergesweakly to P , then F n ( P n ) converges weakly to F ( P ) as n → ∞ . (A2) (I) and (III) hold. (B) The following (B1) and (B2) are equivalent to each other. (B1)
For any sequence {P n } n ∈ N of pyramids and for any pyramid P , if F n ( P n ) converges weakly to F ( P ) , then P n converges weakly to P as n → ∞ . (B2) (I) and (IV) hold. The author obtained in [4] the similar results in the box and concentration topologies toTheorem 1.2. We investigate some properties in the box and concentration topologies likeTheorem 1.3 and obtain the following result.
Theorem 1.5.
Let F n , F : [0 , + ∞ ) → [0 , + ∞ ) , n = 1 , , . . . , be continuous metric preservingfunctions. Then the following (1) – (3) are equivalent to each other. (1) (I) , (II) , and (IV) hold. ONVERGENCE OF METRIC TRANSFORMED SPACES 4 (2)
For any sequence { X n } n ∈ N of metric measure spaces and for any metric measure space X , if F n ( X n ) (cid:3) -converges to F ( X ) , then X n (cid:3) -converges to X as n → ∞ . (3) For any sequence { X n } n ∈ N of metric measure spaces and for any metric measure space X , if F n ( X n ) concentrates to F ( X ) , then X n concentrates to X as n → ∞ .Remark . The results obtained in this paper and in [4] are summarized in the followingtable (see
Table
Table 1 “ X n (cid:3) −−→ X ⇒ F n ( X n ) (cid:3) −−→ F ( X ) ” “ F n ( X n ) (cid:3) −−→ F ( X ) ⇒ X n (cid:3) −−→ X ”iff (I) holds. iff (I), (II), and (IV) hold.“ X n conc −−→ X ⇒ F n ( X n ) conc −−→ F ( X ) ” “ F n ( X n ) conc −−→ F ( X ) ⇒ X n conc −−→ X ”iff (I) and (II) hold. iff (I), (II), and (IV) hold.Assume that F is nondecreasing. Assume that F is nondecreasing.“ X n weak −−→ P ⇒ F n ( X n ) weak −−→ F ( P ) ” “ F n ( X n ) weak −−→ F ( P ) ⇒ X n weak −−→ P ”iff (I), (II), and (III) hold. iff (I), (II), and (IV) hold.Assume that all F n are nondecreasing. Assume that all F n are nondecreasing.“ P n weak −−→ P ⇒ F n ( P n ) weak −−→ F ( P ) ” “ F n ( P n ) weak −−→ F ( P ) ⇒ P n weak −−→ P ”iff (I) and (III) hold. iff (I) and (IV) hold.The top left result is in [4, Corollary 4.4] and the second result on the left-side is in[4, Theorem 1.4] (and see Proposition 4.2 in this paper). Application.
As one of the most important applications of Theorem 1.3 (or Corollary 1.4),we obtain the weak convergence of spheres and projective spaces with the standard Riemann-ian distances.Let S n ( r ) be the n -dimensional sphere in R n +1 centered at the origin and of radius r > S n ( r ) with the standard Riemannian distance function and the normalized volumemeasure. Let F = R , C , or H , where H is the algebra of quaternions, and let d := dim R F .We consider the Hopf quotient F P n ( r ) := S d ( n +1) − ( r ) /U F (1) , where U F (1) := { t ∈ F | k t k = 1 } . This is topologically an n -dimensional projective spaceover F . We equip F P n ( r ) with the quotient metric measure structure of S n ( r ). If F = C ,then the distance function on C P n ( r ) coincides with that induced from the Fubini-Studymetric scaled with factor r . Theorem 1.7.
Let { r n } ∞ n =1 be a given sequence of positive real numbers, and let λ n := r n / √ n ( resp. λ n := r n / √ dn ) . As n → ∞ , S n ( r n ) ( resp. F P n ( r n )) converges weakly to the infinite-dimensional Gaussian space P Γ ∞ λ ( resp. the Hopf quotient P Γ ∞ λ /U F (1) of P Γ ∞ λ ) if and onlyif λ n converges to a positive real number λ . We refer to Subsection 6.1 for the definitions of the infinite-dimensional Gaussian space P Γ ∞ λ and its Hopf quotient P Γ ∞ λ /U F (1). We remark that if the distance functions of S n ( r n ) ONVERGENCE OF METRIC TRANSFORMED SPACES 5 and
F P n ( r n ) are induced from the restriction of the Euclidean distance respectively, the weakconvergence of S n ( r n ) and F P n ( r n ) has been obtained by Shioya [13, 14] and Shioya-Takatsu[15] (see Theorem 6.2). Acknowledgement.
The author would like to thank Professor Takashi Shioya, Professor TakumiYokota, and Professor Ryunosuke Ozawa for their comments and encouragement.2.
Preliminaries
In this section, we describe the definitions and some properties of metric measure space,the box distance, the observable distance, pyramid, and the weak topology. We use most ofthese notions along [13]. As for more details, we refer to [13] and [3, Chapter 3
12 + ].2.1.
Metric measure spaces.
Let (
X, d X ) be a complete separable metric space and m X aBorel probability measure on X . We call the triple ( X, d X , m X ) a metric measure space , oran mm-space for short. We sometimes say that X is an mm-space, in which case the metricand the measure of X are respectively indicated by d X and m X . Definition 2.1 (mm-Isomorphism) . Two mm-spaces X and Y are said to be mm-isomorphic to each other if there exists an isometry f : supp m X → supp m Y such that f ∗ m X = m Y ,where f ∗ m X is the push-forward measure of m X by f . Such an isometry f is called an mm-isomorphism . Denote by X the set of mm-isomorphism classes of mm-spaces.Note that an mm-space X is mm-isomorphic to (supp m X , d X , m X ). We assume that anmm-space X satisfies X = supp m X unless otherwise stated. Definition 2.2 (Lipschitz order) . Let X and Y be two mm-spaces. We say that X ( Lipschitz ) dominates Y and write Y ≺ X if there exists a 1-Lipschitz map f : X → Y satisfying f ∗ m X = m Y . We call the relation ≺ on X the Lipschitz order .The Lipschitz order ≺ is a partial order relation on X .2.2. Box distance and observable distance.
For a subset A of a metric space ( X, d X )and for a real number r >
0, we set U r ( A ) := { x ∈ X | d X ( x, A ) < r } , where d X ( x, A ) := inf a ∈ A d X ( x, a ). We sometimes write U d X r ( A ) if we pay attention to themetric d X . Definition 2.3 (Prokhorov distance) . The
Prokhorov distance d P ( µ, ν ) between two Borelprobability measures µ and ν on a metric space X is defined to be the infimum of ε > µ ( U ε ( A )) ≥ ν ( A ) − ε for any Borel subset A ⊂ X . We sometimes write d d X P ( µ, ν ) if we pay attention to the metric d X . ONVERGENCE OF METRIC TRANSFORMED SPACES 6
The Prokhorov metric d P is a metrization of the weak convergence of Borel probabilitymeasures on X provided that X is a separable metric space. Note that if a map f : X → Y between two metric spaces X and Y is 1-Lipschitz, then we have(2.1) d P ( f ∗ µ, f ∗ ν ) ≤ d P ( µ, ν )for any two Borel probability measures µ and ν on X . Definition 2.4 (Ky Fan metric) . Let (
X, µ ) be a measure space and (
Y, d Y ) a metric space.For two µ -measurable maps f, g : X → Y , we define d µ KF ( f, g ) to be the infimum of ε ≥ µ ( { x ∈ X | d Y ( f ( x ) , g ( x )) > ε } ) ≤ ε. The function d µ KF is a metric on the set of µ -measurable maps from X to Y by identifying twomaps if they are equal to each other µ -almost everywhere. We call d µ KF the Ky Fan metric . Lemma 2.5 ([13, Lemma 1.26]) . Let X be a topological space with a Borel probability measure µ and Y a metric space. For any two Borel measurable maps f, g : X → Y , we have d P ( f ∗ µ, g ∗ µ ) ≤ d µ KF ( f, g ) . Definition 2.6 (Parameter) . Let I := [0 ,
1) and let X be an mm-space. A map ϕ : I → X is called a parameter of X if ϕ is a Borel measurable map such that ϕ ∗ L = m X , where L is the one-dimensional Lebesgue measure on I .Note that any mm-space has a parameter (see [13, Lemma 4.2]). Definition 2.7 (Box distance) . We define the box distance (cid:3) ( X, Y ) between two mm-spaces X and Y to be the infimum of ε ≥ ϕ : I → X , ψ : I → Y , and a Borel subset I ⊂ I with L ( I ) ≥ − ε such that | d X ( ϕ ( s ) , ϕ ( t )) − d Y ( ψ ( s ) , ψ ( t )) | ≤ ε for any s, t ∈ I . Theorem 2.8 ([13, Theorem 4.10]) . The box distance function (cid:3) is a complete separablemetric on X . Lemma 2.9 ([13, Proposition 4.12]) . Let X be a complete separable metric space. For anytwo Borel probability measures µ and ν on X , we have (cid:3) (( X, µ ) , ( X, ν )) ≤ d P ( µ, ν ) . The following notion gives one of the conditions that are equivalent to the box convergence.
Definition 2.10 ( ε -mm-Isomorphism) . Let X and Y be two mm-spaces and f : X → Y aBorel measurable map. Let ε ≥ f is an ε -mm-isomorphism if there exists a Borel subset X ⊂ X such that(1) m X ( X ) ≥ − ε ,(2) | d X ( x, y ) − d Y ( f ( x ) , f ( y )) | ≤ ε for any x, y ∈ X ,(3) d P ( f ∗ m X , m Y ) ≤ ε .We call X a nonexceptional domain of f . ONVERGENCE OF METRIC TRANSFORMED SPACES 7
It is easy to see that, for a 0-mm-isomorphism f : X → Y , there is an mm-isomorphismˆ f : X → Y that is equal to f m X -a.e. on X . Lemma 2.11 ([13, Lemma 4.22]) . Let X and Y be two mm-spaces. (1) If there exists an ε -mm-isomorphism f : X → Y , then (cid:3) ( X, Y ) ≤ ε . (2) If (cid:3) ( X, Y ) < ε , then there exists a ε -mm-isomorphism f : X → Y . Given an mm-space X and a parameter ϕ : I → X of X , we set(2.2) ϕ ∗ L ip ( X ) := { f ◦ ϕ | f : X → R is 1-Lipschitz } . Note that ϕ ∗ L ip ( X ) consists of Borel measurable functions on I . Definition 2.12 (Observable distance) . We define the observable distance d conc ( X, Y ) be-tween two mm-spaces X and Y by d conc ( X, Y ) := inf ϕ,ψ d H ( ϕ ∗ L ip ( X ) , ψ ∗ L ip ( Y )) , where ϕ : I → X and ψ : I → Y run over all parameters of X and Y respectively, and d H isthe Hausdorff distance with respect to the metric d L KF . We say that a sequence { X n } n ∈ N ofmm-spaces concentrates to an mm-space X if X n d conc -converges to X as n → ∞ . Proposition 2.13 ([13, Proposition 5.5]) . For any two mm-spaces X and Y , we have d conc ( X, Y ) ≤ (cid:3) ( X, Y ) . Theorem 2.14 ([13, Theorem 5.13]) . The observable distance function d conc is a metric on X . The basic lemmas used in this paper are listed as follows.
Lemma 2.15 ([13, Corollary 4.48]) . For any mm-space X , there exist -Lipschitz maps Φ N : X → ( R N , k · k ∞ ) , N = 1 , , . . . , such that lim N →∞ (cid:3) ( X, ( R N , k · k ∞ , Φ N ∗ m X )) = 0 , where k x − y k ∞ := max ≤ i ≤ N | x i − y i | for any x, y ∈ R N . Definition 2.16 (1-Lipschitz up to an additive error) . Let X be an mm-space and Y be ametric space. A map f : X → Y is said to be 1- Lipschitz up to ( an additive error ) ε ≥ X ⊂ X such that(1) m X ( X ) ≥ − ε ,(2) d Y ( f ( x ) , f ( x ′ )) ≤ d X ( x, x ′ ) + ε for any x, x ′ ∈ X .We call X a nonexceptional domain of f . Lemma 2.17 ([13, Lemma 5.4]) . If a Borel measurable function f : X → ( R N , k · k ∞ ) onan mm-space X is -Lipschitz up to an additive error ε ≥ , then there exists a -Lipschitzfunction ˜ f : X → ( R N , k · k ∞ ) such that d m X KF ( f, ˜ f ) ≤ ε. Theorem 2.18 ([13, Theorem 4.35]) . Let X , Y , X n , and Y n be mm-spaces, n = 1 , , . . . . If X n and Y n (cid:3) -converge to X and Y respectively as n → ∞ and if X n ≺ Y n for any n , then X ≺ Y . ONVERGENCE OF METRIC TRANSFORMED SPACES 8
Lemma 2.19 ([13, Lemma 6.10]) . (1) If a sequence { X n } n ∈ N of mm-spaces (cid:3) -convergesto an mm-space X and if X dominates an mm-space Y , then there exists a sequence { Y n } n ∈ N of mm-spaces (cid:3) -converging to Y such that X n dominates Y n for each n . (2) If two sequences { X n } n ∈ N and { Y n } n ∈ N of mm-spaces (cid:3) -converge and if X n and Y n are both dominated by an mm-space ˜ Z n for each n , then there exists a sequence of mm-spaces Z n such that X n , Y n ≺ Z n ≺ ˜ Z n and { Z n } n ∈ N has a (cid:3) -convergent subsequence. Pyramid.Definition 2.20 (Pyramid) . A subset
P ⊂ X is called a pyramid if it satisfies the following(1) – (3).(1) If X ∈ P and if Y ≺ X , then Y ∈ P .(2) For any X, X ′ ∈ P , there exists Y ∈ P such that X ≺ Y and X ′ ≺ Y .(3) P is nonempty and (cid:3) -closed.We denote the set of pyramids by Π. Note that Gromov’s definition of a pyramid is only by(1) and (2). The condition (3) is added in [13] for the Hausdorff property of Π.For an mm-space X , we define P X := { X ′ ∈ X | X ′ ≺ X } , which is a pyramid. We call P X the pyramid associated with X .We observe that X ≺ Y if and only if P X ⊂ P Y . Definition 2.21 (Weak convergence) . Let P , P n ∈ Π, n = 1 , , . . . . We say that P n convergesweakly to P as n → ∞ if the following (1) and (2) are both satisfied.(1) For any mm-space X ∈ P , we havelim n →∞ (cid:3) ( X, P n ) = 0 . (2) For any mm-space X ∈ X \ P , we havelim inf n →∞ (cid:3) ( X, P n ) > . Theorem 2.22.
There exists a metric, denoted by ρ , on Π such that the following (1) – (4) hold. (1) ρ is compatible with weak convergence. (2) The map ι : X ∋ X
7→ P X ∈ Π is a -Lipschitz topological embedding map withrespect to d conc and ρ . (3) Π is ρ -compact. (4) ι ( X ) is ρ -dense in Π . In particular, (Π , ρ ) is a compactification of ( X , d conc ). We often identify X with P X ,and we say that a sequence of mm-spaces converges weakly to a pyramid if the associatedpyramid converges weakly. Lemma 2.23 ([13, Lemma 7.14]) . For any pyramid P , there exists a sequence { Y m } m ∈ N ofmm-spaces such that Y ≺ Y ≺ · · · ≺ Y m ≺ · · · and ∞ [ m =1 P Y m (cid:3) = P . ONVERGENCE OF METRIC TRANSFORMED SPACES 9
Such a sequence { Y m } m ∈ N is called an approximation of P . We see that Y m convergesweakly to P as m → ∞ and that Y m ∈ P for all m . Lemma 2.24 (cf. [3, 3 .15.]) . Let P be a pyramid. The following (1) and (2) are equivalentto each other. (1) P ∈ ι ( X ) , i.e., there exists an mm-space X such that P = P X . (2) P is (cid:3) -compact.Proof. We first prove ‘(1) ⇒ (2)’. Take any mm-space X and prove that P X is (cid:3) -compact.We take any real number ε >
0. By [13, Lemma 4.28], it is sufficient to prove that thereexists a real number ∆( ε ) > Y ∈ P X we have a finite net N ⊂ Y suchthat m Y ( U ε ( N )) ≥ − ε, N ≤ ∆( ε ) , and diam N ≤ ∆( ε ) . We find a finite net
N ⊂ X with m X ( U ε ( N )) ≥ − ε . Note that the existence of such N follows from the separability of X . We define∆( ε ) := max { N , diam N } . Take any mm-space Y ∈ P X and fix it. There exists a 1-Lipschitz map f : X → Y such that f ∗ m X = m Y . Then the finite net f ( N ) of Y satisfies f ( N ) ≤ N ≤ ∆( ε ) , diam f ( N ) ≤ diam N ≤ ∆( ε ) ,m Y ( U ε ( f ( N ))) ≥ m X ( U ε ( N )) ≥ − ε. Thus P X is (cid:3) -compact.We next prove ‘(2) ⇒ (1)’. Let { Y m } m ∈ N be an approximation of P . Note that Y m converges weakly to P . Since P is (cid:3) -compact, the sequence { Y m } m ∈ N has a (cid:3) -convergentsubsequence. Let X be a limit mm-space of a (cid:3) -convergent subsequence of { Y m } m ∈ N . Since a (cid:3) -convergence becomes a weak convergence, we have P = P X . This completes the proof. (cid:3) Metric preserving functions
In this section, we recall some properties of metric preserving function. The notion ofmetric preserving functions was discovered in 1930s, and the study of these functions hasbeen deepened. We refer to [1] for a survey of results on metric preserving functions.3.1.
Metric preserving functions.Definition 3.1 (Metric preserving function) . A function F : [0 , + ∞ ) → [0 , + ∞ ) is called a metric preserving function provided that for any metric space ( X, d X ), the function F ◦ d X is a metric on X . Lemma 3.2 ([1, Propositions 2.1, 2.3]) . Let F : [0 , + ∞ ) → [0 , + ∞ ) be a function. Then thefollowing (1) and (2) hold. (1) If F is a metric preserving function, then F is subadditive ( i.e., F ( s + t ) ≤ F ( s ) + F ( t ) for any s, t ≥ and F − (0) = { } . ONVERGENCE OF METRIC TRANSFORMED SPACES 10 (2) If F is subadditive and nondecreasing and fulfills F − (0) = { } , then F is a metricpreserving function. In particular, if F is a concave function with F − (0) = { } , then F is a metric preserving function.Remark . There are many examples of metric preserving functions that are not nonde-creasing. For example, F ( s ) := s if s ∈ [0 , , − s if s ∈ [2 , , s ∈ [3 , + ∞ ) ,G ( s ) := s if s ∈ [0 , , s + sin ( s − s if s ∈ [1 , + ∞ ) . Proposition 3.4 (cf. [1, Propositions 2.6]) . Let F : [0 , + ∞ ) → [0 , + ∞ ) be a metric preserv-ing function. Then, for any s, t ≥ , we have (1) | F ( s ) − F ( t ) | ≤ F ( | s − t | ) , (2) F ( s ) ≤ F ( t ) if s ≤ t . Theorem 3.5 ([1, Theorem 3.4]) . Let F : [0 , + ∞ ) → [0 , + ∞ ) be a metric preserving func-tion. Then the following conditions are equivalent to each other. (1) F is continuous. (2) F is continuous at . (3) F is uniformly continuous. (4) For any metric space ( X, d X ) , the topologies induced by d X and F ◦ d X coincide witheach other. Note that if F is discontinuous, then F ◦ d X gives the discrete topology on X . Proposition 3.6 (cf. [4, Proposition 3.10]) . Let F : [0 , + ∞ ) → [0 , + ∞ ) be a metric preserv-ing function. If a metric space ( X, d X ) is complete, then so is ( X, F ◦ d X ) . Convergence of metric preserving functions.
In this subsection, we describe someproperties for a sequence of metric preserving functions. In particular, we show some condi-tions that are equivalent to (I), (II), and (III) respectively.
Lemma 3.7.
Let F n , F : [0 , + ∞ ) → [0 , + ∞ ) , n = 1 , , . . . , be metric preserving functions.If F is continuous and if F n converges pointwise to F , then F n converges uniformly to F oncompact sets.Proof. We take any compact set K ⊂ [0 , + ∞ ) and any real number ε >
0. Let us prove that(3.1) sup s ∈ K | F n ( s ) − F ( s ) | ≤ ε holds for every sufficiently large n . By the continuity of F , there exists a real number δ > F ( δ ) ≤ ε . By the compactness of K , we find finite points { s i } ki =1 in K such that K ⊂ k [ i =1 U δ ( s i ) . ONVERGENCE OF METRIC TRANSFORMED SPACES 11
Let N ∈ N be a number such thatmax i =1 ,...,k | F n ( s i ) − F ( s i ) | ≤ ε and | F n ( δ ) − F ( δ ) | ≤ ε hold for all n ≥ N . Given a fixed point s ∈ K , we find i ∈ { , . . . , k } such that s ∈ U δ ( s i ).By Proposition 3.4, we have | F n ( s ) − F ( s ) |≤ | F n ( s i ) − F ( s i ) | + F n ( | s − s i | ) + F ( | s − s i | ) ≤ | F n ( s i ) − F ( s i ) | + 2 F n ( δ ) + 2 F ( δ ) ≤ | F n ( s i ) − F ( s i ) | + 2 | F n ( δ ) − F ( δ ) | + 4 F ( δ ) ≤ ε for every n ≥ N . Thus we obtain (3.1). This completes the proof. (cid:3) Given a function F : [0 , + ∞ ) → [0 , + ∞ ), we set I F ( s ) := F ( s ) − inf s ≤ s ′ F ( s ′ ) , s ∈ [0 , + ∞ ) . Note that I F ≥ I F ≡ F is nondecreasing. Lemma 3.8.
Let F n : [0 , + ∞ ) → [0 , + ∞ ) be functions, n = 1 , , . . . . Assume that F n con-verges uniformly to a continuous function F on compact sets. Then the following (1) – (3) are equivalent to each other. (1) For any s > , lim n →∞ I F n ( s ) = 0 holds ( i.e., (II) holds ) . (2) For any
D > , lim n →∞ sup s ∈ [0 ,D ] I F n ( s ) = 0 holds. (3) F is nondecreasing and, for any sequence s n → ∞ , lim inf n →∞ F n ( s n ) ≥ sup F (cid:16) = lim n →∞ F ( s n ) (cid:17) holds.Proof. ‘(2) ⇒ (1)’ is obvious. We verify ‘(1) ⇒ (3)’ and ‘(3) ⇒ (2)’.Assume (1). Take any two real numbers s, s ′ with 0 ≤ s ≤ s ′ . Then F ( s ) = lim n →∞ F n ( s ) ≤ lim n →∞ ( F n ( s ′ ) + I F n ( s )) = F ( s ′ ) , which implies that F is nondecreasing. We take any sequence s n → ∞ and any real number s ≥
0. For every sufficiently large n , since s ≤ s n , we see that F n ( s ) ≤ F n ( s n ) + I F n ( s ) . Thus we have F ( s ) = lim n →∞ F n ( s ) ≤ lim inf n →∞ ( F n ( s n ) + I F n ( s )) = lim inf n →∞ F n ( s n ) , ONVERGENCE OF METRIC TRANSFORMED SPACES 12 which implies sup F ≤ lim inf n →∞ F n ( s n ) . The proof of ‘(1) ⇒ (3)’ is completed.We next prove ‘(3) ⇒ (2)’. Suppose that (2) does not hold. Then there exists D > η := lim sup n →∞ sup s ∈ [0 ,D ] I F n ( s ) > . Taking a subsequence of n , we can assume that sup s ∈ [0 ,D ] I F n ( s ) → η . Thus, for everysufficiently large n , we have sup s ∈ [0 ,D ] I F n ( s ) > η . Then, there exist two real numbers s n , s ′ n with 0 < s n ≤ min { s ′ n , D } such that F n ( s n ) > F n ( s ′ n ) + η . Taking a subsequence again, we are able to assume that at least one of the following twosituations occurs. • s n and s ′ n converge to real numbers s ∞ and s ′∞ respectively. • s n converges to a real number s ∞ and s ′ n diverges to + ∞ .If the first situation occurs, then we have s ∞ ≤ s ′∞ and F ( s ∞ ) ≥ F ( s ′∞ ) + η . In fact, since F n converges uniformly to F on compact sets, we have F n ( s n ) → F ( s ) if s n → s .However this contradicts the monotonicity of F . If the second situation occurs, then we have F ( s ∞ ) ≥ lim inf n →∞ F n ( s ′ n ) + η , which contradicts lim inf n →∞ F n ( s ′ n ) ≥ sup F . Therefore we obtain ‘(3) ⇒ (2)’.The proof of this lemma is completed. (cid:3) Remark . Under the setting of Lemma 3.8, we consider the following other conditions.(i) The functions F n are nondecreasing for all n ∈ N .(ii) lim n →∞ sup s ≥ I F n ( s ) = 0.(iii) The function F is nondecreasing.It is easy to see that ‘(i) ⇒ (ii) ⇒ (II) ⇒ (iii)’. On the other hand, the converse of each ofthese implications does not hold, even in the class of metric preserving functions. In fact, weshow the following examples. We define functions F n , F n , and F n , n = 1 , , . . . , by F n ( s ) := s if s ∈ [0 , , − s if s ∈ [2 , n − ) , − n − if s ∈ [2 + n − , + ∞ ) . ONVERGENCE OF METRIC TRANSFORMED SPACES 13 F n ( s ) := s if s ∈ [0 , , s ∈ [2 , n + 2) ,s − n if s ∈ [ n + 2 , n + 3) ,n + 6 − s if s ∈ [ n + 3 , n + 4) , s ∈ [ n + 4 , + ∞ ) .F n ( s ) := s if s ∈ [0 , , s ∈ [2 , n + 2) ,n + 4 − s if s ∈ [ n + 2 , n + 3) , s ∈ [ n + 3 , + ∞ ) . These functions are continuous metric preserving functions and converge to the concavefunction min { s, } as n → ∞ . { F n } , { F n } , and { F n } are counterexamples of ‘(ii) ⇒ (i)’,‘(II) ⇒ (ii)’, and ‘(iii) ⇒ (II)’ respectively. Lemma 3.10.
Let F n : [0 , + ∞ ) → [0 , + ∞ ) be functions, n = 1 , , . . . . Assume that F n converges uniformly to a continuous function F on compact sets. Then the following (1) – (3) are equivalent to each other. (1) lim sup n →∞ sup F n ≤ sup F holds ( i.e., (III) holds ) . (2) lim sup n →∞ sup F n = sup F holds. (3) For any sequence s n → ∞ , lim sup n →∞ F n ( s n ) ≤ sup F holds.Proof. We first verify that lim inf n →∞ sup F n ≥ sup F is always true. For any s ≥
0, we have F ( s ) = lim n →∞ F n ( s ) ≤ lim inf n →∞ sup F n , which implies sup F ≤ lim inf n →∞ sup F n . Therefore we obtain ‘(1) ⇔ (2)’. Moreover, ‘(1) ⇒ (3)’ is trivial.We verify ‘(3) ⇒ (1)’. We first assume that sup F n < + ∞ for all n . We take any realnumber ε >
0. There exists a sequence { s n } of positive real numbers such thatsup F n ≤ F n ( s n ) + ε for every n . Taking a subsequence, we can assume that { s n } converges to a real number s ∞ or it diverges to infinity. If s n → s ∞ , then we havelim sup n →∞ sup F n ≤ F ( s ∞ ) + ε ≤ sup F + ε. ONVERGENCE OF METRIC TRANSFORMED SPACES 14 If s n → + ∞ , then we havelim sup n →∞ sup F n ≤ lim sup n →∞ F n ( s n ) + ε ≤ sup F + ε. Thus, as ε →
0, we obtain (1). In the case that sup F n i = + ∞ for some subsequence { n i } ,for any real number M >
0, there exists a sequence { s i } of positive real numbers such that F n i ( s i ) > M for every i . In the same discussion as above, taking a subsequence of { n i } , we obtain M < lim sup i →∞ F n i ( s i ) ≤ sup F, which implies sup F = + ∞ . Thus we obtain (1) in general.The proof of this lemma is completed. (cid:3) Proposition 3.11.
Let F n , F : [0 , + ∞ ) → [0 , + ∞ ) , n = 1 , , . . . , be continuous metric pre-serving functions. Assume that, for any sequence { s n } ⊂ [0 , + ∞ ) and any s > , if F n ( s n ) converges to F ( s ) , then s n converges to s . Then F n converges pointwise to F .Proof. We take any s > F ( s ) ≤ lim inf n →∞ F n ( s ) . We set α := lim inf n →∞ F n ( s ) and suppose that α < F ( s ). There exists a subsequence { n i } i of n such that F n i ( s ) → α as i → ∞ . For each n , if α < F n ( s ), then there exists a realnumber s ′ n > F n ( s ′ n ) = α by the intermediate value theorem. We set a sequence s n := (cid:26) s ′ n if n = n i and if α < F n ( s ) ,s otherwise.Taking the definition of α into account, we see that F n ( s n ) converges to α as n → ∞ . By α < F ( s ) and by the intermediate value theorem, there exists a real number β > α = F ( β ). Thus, by the assumption of this proposition, we have s n → β as n → ∞ . Since { s n } has a subsequence consisting only of s , we have β = s , which contradicts α < F ( s ).Thus we have (3.2).We next prove(3.3) lim sup n →∞ F n ( s ) ≤ F ( s ) . Suppose that F n ( s ) > F ( s ) + η for every sufficiently large n and for some real number η >
0. For every sufficiently large n , there exists a real number s n > s n ≤ s and F n ( s n ) = F ( s ). By the assumption of this proposition, s n converges to s . Since F n is ametric preserving function, for every sufficiently large n , we have0 < η < F n ( s ) − F ( s ) = F n ( s ) − F n ( s n ) ≤ F n ( s − s n ) . Let η ′ := min { F ( s ) , η } >
0. For every sufficiently large n , there exists a real number t n suchthat t n ≤ s − s n and F n ( t n ) = η ′ . We see that t n converges to 0. On the other hand, since areal number t > F ( t ) = η ′ is also found, it is follows from the assumption that t n converges to t . This is a contradiction. Thus we obtain (3.3). The proof is completed. (cid:3) ONVERGENCE OF METRIC TRANSFORMED SPACES 15 Weak convergence of metric transformed pyramids
The goal in this section is to prove Theorem 1.2 and Theorem 1.3.We review the definitions of F ( X ) and F ( P ) in Definition 1.1. Let F : [0 , + ∞ ) → [0 , + ∞ )be a continuous metric preserving function. Given an mm-space X and a pyramid P , wedefine F ( X ) := ( X, F ◦ d X , m X ) and F ( P ) := [ X ∈P P F ( X ) (cid:3) . By Theorem 3.5 and Proposition 3.6, F ( X ) is an mm-space for a given mm-space X . Proposition 4.1.
Let F : [0 , + ∞ ) → [0 , + ∞ ) be a continuous metric preserving function. If F is nondecreasing, then F ( P ) is a pyramid for any pyramid P and F ( P X ) = P F ( X ) holdsfor any mm-space X .Proof. Let P be a pyramid. We verify that F ( P ) is a pyramid. It is obvious that F ( P ) isnonempty and (cid:3) -closed.We check the condition (1) of Definition 2.20. Assume Y ∈ F ( P ) and Y ′ ≺ Y . By thedefinition of F ( P ), there exist mm-spaces X n ∈ P and Y n ∈ X , n = 1 , , . . . , such that F ( X n )dominates Y n for every n and (cid:3) ( Y n , Y ) → n → ∞ . By Lemma 2.19 (1), there exists asequence { Y ′ n } n ∈ N of mm-spaces such that Y n dominates Y ′ n for every n and (cid:3) ( Y ′ n , Y ′ ) → n → ∞ . Since F ( X n ) dominates Y ′ n (i.e., Y ′ n ∈ P F ( X n )) for every n , we have Y ′ ∈ F ( P ).We next check the condition (2) of Definition 2.20. Take any two mm-spaces Y, Y ′ ∈ F ( P ).By the definition of F ( P ), there exist mm-spaces X n , X ′ n ∈ P and Y n , Y ′ n ∈ X , n = 1 , , . . . ,such that F ( X n ) (resp. F ( X ′ n )) dominates Y n (resp. Y ′ n ) for every n and (cid:3) ( Y n , Y ) → (cid:3) ( Y ′ n , Y ′ ) → n → ∞ . By X n , X ′ n ∈ P , there exists e X n ∈ P such that e X n dominatesboth X n and X ′ n . Since F is nondecreasing, we see that F ( e X n ) dominates both F ( X n ) and F ( X ′ n ), which implies that F ( e X n ) dominates both Y n and Y ′ n . By Lemma 2.19 (2), there existsa sequence { Z n } n ∈ N of mm-spaces such that Y n , Y ′ n ≺ Z n ≺ F ( e X n ) for every n and { Z n } n ∈ N has a (cid:3) -convergent subsequence. Let Z be a limit space of (cid:3) -convergent subsequence of { Z n } n ∈ N . Since { Z n } n ∈ N ⊂ F ( P ), we see that Z ∈ F ( P ) and, by Theorem 2.18, we have Y, Y ′ ≺ Z . Thus F ( P ) satisfies the condition (2), so that F ( P ) is a pyramid.We prove that F ( P X ) = P F ( X ) for any mm-space X . We take an mm-space X and fixit. Since X ∈ P X , we have F ( P X ) ⊃ P F ( X ). Since F is nondecreasing, if Y ∈ P X , then P F ( Y ) ⊂ P F ( X ), which leads to F ( P X ) ⊂ P F ( X ). This completes the proof. (cid:3) Proof of Theorem 1.2.Proposition 4.2.
Let F n , F : [0 , + ∞ ) → [0 , + ∞ ) , n = 1 , , . . . , be continuous metric pre-serving functions. Assume that, for any mm-space X , the metric transformed space F n ( X ) concentrates to F ( X ) as n → ∞ . Then, F n converges pointwise to F as n → ∞ .Proof. Take any real number s > X is defined as X := ( { , s } , | · | , δ ,s ) , where δ ,s := 1 / δ + 1 / δ s and δ x is the Dirac probability measure at x . By the assumption, F n ( X ) concentrates to F ( X ) as n → ∞ . Note that F n ( X ) is mm-isomorphic to the mm-space( { , F n ( s ) } , | · | , δ ,F n ( s ) ) . ONVERGENCE OF METRIC TRANSFORMED SPACES 16
Suppose that F n ( s ) diverges to infinity as n → ∞ . It is easy to see that F n ( X ) convergesweakly to the pyramid P := { ( { , t } , | · | , δ ,t ) | t ≥ } . Since P is not (cid:3) -precompact (see [13, Lemma 4.28]), it follows from Lemma 2.24 that P 6 = P F ( X ), which is a contradiction. Thus we can assume that F n ( s ) converges to a nonnegativenumber t . Then, F n ( X ) (cid:3) -converges to ( { , t } , | · | , δ ,t ), so that we have t = F ( s ). Thiscompletes the proof. (cid:3) The following lemma was obtained in [4].
Lemma 4.3 ([4, Claim 5.1]) . Let F n , F : [0 , + ∞ ) → [0 , + ∞ ) , n = 1 , , . . . , be continuousmetric preserving functions. Assume that F n converges pointwise to F as n → ∞ . If (II) does not hold, then there exist a sequence { X n } n ∈ N of mm-spaces and two mm-spaces X , Y such that (1) X n concentrates to X as n → ∞ , (2) F n ( X n ) concentrates to Y as n → ∞ , (3) F ( X ) and Y are not mm-isomorphic to each other.Moreover, if F is increasing, then there exists an mm-space X ′ such that Y = F ( X ′ ) . Proposition 4.4.
Let F n , F : [0 , + ∞ ) → [0 , + ∞ ) , n = 1 , , . . . , be continuous metric pre-serving functions. Assume that F n converges pointwise to F as n → ∞ and that F isnondecreasing. If lim sup n →∞ sup F n > sup F, then there exists a sequence { X n } n ∈ N of mm-spaces converging weakly to a pyramid P suchthat F n ( X n ) does not converge weakly to F ( P ) .Proof. Assume that lim sup n →∞ sup F n > sup F . We define a pyramid P := { ( { , s } , | · | , δ ,s ) | s ≥ } ⊂ X , where the notation is same as in the proof of Proposition 4.2. We set α := lim sup n →∞ sup F n and η := (cid:26) α − sup F if α < + ∞ , α = + ∞ . Note that η >
0. There exists a subsequence { n i } ⊂ { n } such thatsup F n i > sup F + η. For each i , there exists s i ∈ [0 , + ∞ ) such that F n i ( s i ) > sup F + η. We see that s i → ∞ as i → ∞ . In fact, if s := lim inf i →∞ s i < + ∞ , then, by Lemma 3.7, we have F ( s ) ≥ sup F + η, which is a contradiction. We define mm-spaces X i , i = 1 , , . . . , by X i := ( { , s i } , | · | , δ ,s i ) , ONVERGENCE OF METRIC TRANSFORMED SPACES 17 and then it follows from s i → ∞ that X i converges weakly to P as i → ∞ . We prove that F n i ( X i ) does not converge weakly to F ( P ). Since sup F + η < F n i ( s i ) for any i , we have( { , sup F + η } , | · | , δ , sup F + η ) ∈ P F n i ( X i ) , which implies lim inf n →∞ (cid:3) (( { , sup F + η } , | · | , δ , sup F + η ) , P F n i ( X i )) = 0 . On the other hand, since diam Y ≤ sup F for any Y ∈ F ( P ), we have( { , sup F + η } , | · | , δ , sup F + η ) F ( P ) . Thus we obtain the conclusion. (cid:3)
Proof of ‘(1) ⇒ (2)’ of Theorem 1.2. This follows from Proposition 4.2, Lemma 4.3, andProposition 4.4 directly. (cid:3)
We prepare some lemmas for the proof of ‘(2) ⇒ (1)’ of Theorem 1.2. Lemma 4.5.
Let F n , F : [0 , + ∞ ) → [0 , + ∞ ) , n = 1 , , . . . , be continuous metric preservingfunctions satisfying (I) and (II) . If Borel measurable maps f n : X n → X between mm-spaces X n and X are -Lipschitz up to ε n for some ε n → , and if d d X P ( f n ∗ m X n , m X ) ≤ ε n holds,then there exists a sequence δ n → such that f n is -Lipschitz up to δ n with respect to F n ◦ d X n and F ◦ d X , and d F ◦ d X P ( f n ∗ m X n , m X ) ≤ δ n . Proof.
We take any real number ε >
0. By the inner regularity of m X , there exists a compactset K ⊂ X such that m X ( K ) ≥ − ε . We put D ε := diam K + 2 ε . By the assumptions ofthis lemma, Lemma 3.7, and Lemma 3.8, for any sufficiently large n , • f n : X n → X is 1-Lipschitz up to ε and d P ( f n ∗ m X n , m X ) ≤ ε holds, • | F n ( s ) − F ( s ) | < ε holds for any s ∈ [0 , D ε ], • F n ( s ) ≤ F n ( s ′ ) + ε holds for any s ∈ [0 , D ε ] and for any s ′ ≥ s .Let X ′ n ⊂ X n be a nonexceptional domain of f n : X n → X and let e X n := X ′ n ∩ f − n ( U ε ( K )) . We see that m X n ( ˜ X n ) ≥ m X n ( X ′ n ) + f n ∗ m X n ( U ε ( K )) − ≥ − ε. For any x, x ′ ∈ e X n , we have F ( d X ( f n ( x ) , f n ( x ′ ))) ≤ F n ( d X ( f n ( x ) , f n ( x ′ ))) + ε ≤ F n ( d X n ( x, x ′ ) + ε ) + 2 ε ≤ F n ( d X n ( x, x ′ )) + F n ( ε ) + 2 ε ≤ F n ( d X n ( x, x ′ )) + F ( ε ) + 3 ε, where the first and second inequalities follow fromdiam f n ( e X n ) ≤ D ε . Thus, for any sufficiently large n , the map f n : F n ( X n ) → F ( X ) is 1-Lipschitz up to F ( ε )+3 ε .We next prove d F ◦ d X P ( f n ∗ m X n , m X ) ≤ max { ε, F ( ε ) } . For any subset A ⊂ X and any realnumber η >
0, we have U d X ε ( A ) ⊂ U F ◦ d X F ( ε )+ η ( A ) . ONVERGENCE OF METRIC TRANSFORMED SPACES 18
In fact, taking any point x ∈ U d X ε ( A ), it holds that d X ( x, A ) < ε , which implies that F ( d X ( x, A )) ≤ F ( ε ). Combining d d X P ( f n ∗ m X n , m X ) ≤ ε and this leads to m X ( A ) ≤ f n ∗ m X n ( U d X ε ( A )) + ε ≤ f n ∗ m X n ( U F ◦ d X F ( ε )+ η ( A )) + ε, which implies that d F ◦ d X P ( f n ∗ m X n , m X ) ≤ max { ε, F ( ε ) + η } . As η →
0, we obtain d F ◦ d X P ( f n ∗ m X n , m X ) ≤ max { ε, F ( ε ) } . The proof is completed. (cid:3)
Lemma 4.6.
Let X and Y be two mm-spaces and let ε > . (1) If a Borel measurable map f : X → Y is -Lipschitz up to ε and if d P ( f ∗ m X , m Y ) ≤ ε holds, then (cid:3) ( Y, P X ) ≤ ε . (2) If (cid:3) ( Y, P X ) < ε , then there exists a Borel measurable map f : X → Y that is -Lipschitz up to ε and d P ( f ∗ m X , m Y ) ≤ ε holds.Proof. We first prove (1). Let f : X → Y be a map satisfying the assumption. By Lemma2.15, there exist 1-Lipschitz maps Φ N : Y → ( R N , k · k ∞ ), N = 1 , , . . . , such thatlim N →∞ (cid:3) ( Y, ( R N , k · k ∞ , Φ N ∗ m Y )) = 0 . Since the composition Φ N ◦ f : X → ( R N , k · k ∞ ) is 1-Lipschitz up to ε , by Lemma 2.17, thereexists a 1-Lipschitz map Ψ N : X → ( R N , k · k ∞ ) such that d m X KF (Φ N ◦ f, Ψ N ) ≤ ε. Note that ( R N , k · k ∞ , Ψ N ∗ m X ) ∈ P X . Then we have d P (Φ N ∗ m Y , Ψ N ∗ m X ) ≤ d P ((Φ N ◦ f ) ∗ m X , Ψ N ∗ m X ) + d P ((Φ N ◦ f ) ∗ m X , Φ N ∗ m Y ) ≤ d m X KF (Φ N ◦ f, Ψ N ) + d P ( f ∗ m X , m Y ) ≤ ε, where the second inequality follows from Lemma 2.5 and (2.1). Thus we have (cid:3) ( Y, P X ) ≤ (cid:3) ( Y, ( R N , k · k ∞ , Ψ N ∗ m X )) ≤ (cid:3) ( Y, ( R N , k · k ∞ , Φ N ∗ m Y )) + 2 d P (Φ N ∗ m Y , Ψ N ∗ m X ) ≤ (cid:3) ( Y, ( R N , k · k ∞ , Φ N ∗ m Y )) + 4 ε. As N → ∞ , we obtain (cid:3) ( Y, P X ) ≤ ε .We next prove (2). There exists an mm-space X ′ dominated by X such that (cid:3) ( Y, X ′ ) < ε .There exist a 1-Lipschitz map f : X → X ′ with f ∗ m X = m X ′ and a 3 ε -mm-isomorphism g : X ′ → Y by Lemma 2.11. It is easy to see that the composition g ◦ f : X → Y is 1-Lipschitz up to 3 ε and that d P (( g ◦ f ) ∗ m X , m Y ) ≤ ε . This completes the proof. (cid:3) We see that there exists an mm-space X ′ ∈ P X minimizing (cid:3) ( Y, P X ) by Lemma 2.24. Corollary 4.7.
Let P be a pyramid and let Y an mm-space. Assume that, for any ε > ,there exist an mm-space X ε ∈ P and a Borel measurable map f ε : X ε → Y such that f ε is -Lipschitz up to ε and d P ( f ε ∗ m X ε , m Y ) ≤ ε holds. Then Y ∈ P . ONVERGENCE OF METRIC TRANSFORMED SPACES 19
Proof.
Take any ε >
0. There exist an mm-space X ε ∈ P and a map f ε : X ε → Y in theassumption. By Lemma 4.6, we have (cid:3) ( Y, P X ε ) ≤ ε . Since P X ε ⊂ P , we have (cid:3) ( Y, P ) ≤ (cid:3) ( Y, P X ε ) ≤ ε. As ε →
0, we obtain Y ∈ P . (cid:3) We prove ‘(2) ⇒ (1)’ of Theorem 1.2. Let F n , F be continuous metric preserving functionssatisfying (I), (II), and (III). Take any sequence { X n } of mm-spaces and any pyramid P ∈
Πsuch that X n converges weakly to P . Our goal is to prove that F n ( X n ) converges weakly to F ( P ) as n → ∞ . Proposition 4.8.
For any Y ∈ F ( P ) , we have lim n →∞ (cid:3) ( Y, P F n ( X n )) = 0 . Proof of Proposition 4.8.
Take any mm-space Y ∈ F ( P ) and any real number ε >
0. Thereexist two mm-spaces X ′ ∈ P and Y ′ ∈ X such that F ( X ′ ) dominates Y ′ and (cid:3) ( Y ′ , Y ) ≤ ε .Since X n converges weakly to P , we have (cid:3) ( X ′ , P X n ) → n → ∞ . Then, by Lemma 4.6,there exist Borel measurable maps f n : X n → X ′ , n = 1 , , . . . , such that f n is 1-Lipschitz upto ε n and d P ( f n ∗ m X n , m X ′ ) ≤ ε n holds for some ε n →
0. By Lemma 4.5, we see that the map f n : F n ( X n ) → F ( X ′ ) is 1-Lipschitz up to δ n and d F ◦ d X ′ P ( f n ∗ m X n , m X ′ ) ≤ δ n holds for some δ n →
0. Moreover, since F ( X ′ ) dominates Y ′ , there exists a 1-Lipschitz map g : F ( X ′ ) → Y ′ with g ∗ m X ′ = m Y ′ . Since the composition g ◦ f n : F n ( X n ) → Y ′ is also 1-Lipschitz up to δ n and fulfills d P (( g ◦ f n ) ∗ m X n , m Y ′ ) ≤ δ n , we have (cid:3) ( Y ′ , P F n ( X n )) ≤ δ n by Lemma 4.6. Thus we have (cid:3) ( Y, P F n ( X n )) ≤ (cid:3) ( Y, Y ′ ) + (cid:3) ( Y ′ , P F n ( X n )) ≤ ε + 4 δ n . As n → ∞ and ε →
0, we obtain the conclusion. (cid:3)
Proposition 4.9.
If an mm-space Y satisfies lim inf n →∞ (cid:3) ( Y, P F n ( X n )) = 0 , then Y ∈ F ( P ) . The following proof is inspired by the ideas invented first by Ryunosuke Ozawa and TakumiYokota. The author was privately informed of their ideas.
Proof.
Assume that lim inf n →∞ (cid:3) ( Y, P F n ( X n )) = 0. We can assume that Y = {∗} . Choosinga subsequence of n , we can assume that (cid:3) ( Y, P F n ( X n )) → n → ∞ . Then, by Lemma4.6, there exist Borel measurable maps f n : F n ( X n ) → Y and a sequence ε n → f n is 1-Lipschitz up to ε n and d P ( f n ∗ m X n , m Y ) ≤ ε n holds for every n . Let e X n ⊂ X n be anonexceptional domain of f n : F n ( X n ) → Y .We take any sufficiently small real number ε >
0. We find finite many open sets B , . . . , B N in Y such that diam B i < ε and m Y ( B i ) > i ∈ { , . . . , N } and that δ ′ := min ≤ i
Let B := Y \ F Ni =1 B i . For any i = 0 , . . . , N , we take any point y i ∈ B i and fix it. If B = ∅ ,we consider only for i = 1 , . . . , N . An mm-space ˙ Y is defined as˙ Y := ( { y i } Ni =0 , d Y , m ˙ Y ) , where m ˙ Y ( { y i } ) := m Y ( B i ). Note that the natural embedding ι : ˙ Y ∋ y i y i ∈ Y is an ε -mm-isomorphism. Our goal is, by Corollary 4.7, to prove that there exist an mm-space W ε ∈ F ( P ) and a map h ε : W ε → Y such that h ε is 1-Lipschitz up to 4 ε and d P ( h ε ∗ m W ε , m Y ) ≤ ε .For any n ∈ N and for any i = 1 , . . . , N , we define a subset A n,i := f − n ( B i ) ∩ e X n ⊂ X n and define a real number R := inf (cid:26) r > (cid:12)(cid:12)(cid:12)(cid:12) max i,j =1 ,...,N d Y ( y i , y j ) − ε ≤ F ( r ) (cid:27) . The existence of R is discussed as follows. For any i, j = 1 , . . . , N , taking any x n,i ∈ A n,i andany x n,j ∈ A n,j , we have d Y ( y i , y j ) ≤ d Y ( f n ( x n,i ) , f n ( x n,j )) + 2 ε ≤ F n ( d X n ( x n,i , x n,j )) + ε n + 2 ε ≤ sup F n + ε n + 2 ε. By (III), we obtain d Y ( y i , y j ) − ε ≤ sup F , which means that R exists.We define a map Φ n : X n → ( R N , k · k ∞ ) byΦ n ( x ) := (min { d X n ( x, A n,i ) , R } ) Ni =1 , x ∈ X n . Since the measure ν n := Φ n ∗ m X n is supported in the compact set (cid:8) z ∈ R N (cid:12)(cid:12) k z k ∞ ≤ R (cid:9) , asequence { ν n } n ∈ N is tight. Thus the sequence { ν n } has a subsequence converging weakly toa Borel probability measure ν on R N . An mm-space Z is defined as Z := (supp ν, k · k ∞ , ν ) . It follows from ( R N , k · k ∞ , ν n ) ≺ X n that (cid:3) ( Z, P X n ) → n → ∞ . Since X n convergesweakly to P , we have Z ∈ P .Let δ := 12 inf (cid:26) r > (cid:12)(cid:12)(cid:12)(cid:12) δ ′ ≤ F ( r ) (cid:27) > A i := (cid:8) ( z i ) Ni =1 ∈ R N (cid:12)(cid:12) z i = 0 and | z j | ≥ δ for any j = i (cid:9) for any i = 1 , . . . , N . Note that 2 δ ≤ R . In fact, if 2 δ > R , then we have δ ′ > F ( R ) ≥ max i,j =1 ,...,N d Y ( y i , y j ) − ε ≥ δ ′ − ε, which implies that max i,j d Y ( y i , y j ) ≤ ε . This contradicts Y = {∗} . Claim 4.10.
For any i = 1 , . . . , N and for any sufficiently large n , we have Φ n ( A n,i ) ⊂ A i . ONVERGENCE OF METRIC TRANSFORMED SPACES 21
Proof.
Take any i ∈ { , . . . , N } and fix it. Since(Φ n ( x )) i = min { d X n ( x, A n,i ) , R } = 0for any x ∈ A n,i , it is sufficient to prove that(4.1) lim inf n →∞ d X n ( A n,i , A n,j ) ≥ δ for every j = i . In fact, taking 2 δ ≤ R into account, we havelim inf n →∞ inf x ∈ A n,i (Φ n ( x )) j ≥ δ, which implies that inf x ∈ A n,i (Φ n ( x )) j ≥ δ for any sufficiently large n . We prove (4.1). Weput α := lim inf n →∞ d X n ( A n,i , A n,j ). There exists a subsequence { n k } ⊂ { n } such that d X nk ( A n k ,i , A n k ,j ) → α as k → ∞ . We find two sequences { x m } m ⊂ A n k ,i and { x ′ m } m ⊂ A n k ,j such that d X nk ( x m , x ′ m ) → d X nk ( A n k ,i , A n k ,j ) as m → ∞ . Then we have δ ′ ≤ d Y ( B i , B j ) ≤ d Y ( f n ( x m ) , f n ( x ′ m )) ≤ F n k ( d X nk ( x m , x ′ m )) + ε n k → F n k ( d X nk ( A n k ,i , A n k ,j )) + ε n k as m → ∞ for each k . Moreover, by Lemma 3.7, we havelim k →∞ F n k ( d X nk ( A n k ,i , A n k ,j )) = F ( α ) . Combining these implies δ ′ ≤ F ( α ), so that 2 δ ≤ α . Therefore we obtain (4.1). Thiscompletes the proof. (cid:3) We define a subset e Z ⊂ Z by e Z := supp ν ∩ N G i =1 A i and define a map g : F ( Z ) → ˙ Y by g ( z ) := (cid:26) y i if z ∈ A i ,y if z ∈ Z \ e Z. If B = ∅ , then we set g ( z ), for z ∈ Z \ e Z , an arbitrary point of ˙ Y . Claim 4.11.
The map g : F ( Z ) → ˙ Y is -Lipschitz up to ε and fulfills d P ( g ∗ ν, m ˙ Y ) ≤ ε. If we prove this claim, then the composition ι ◦ g : F ( Z ) → Y is 1-Lipschitz up to 4 ε and d P (( ι ◦ g ) ∗ ν, m Y ) ≤ ε holds. Combining this and Corollary 4.7 implies Y ∈ F ( P ). Proof of Claim 4.11.
We first prove d P ( g ∗ ν, m ˙ Y ) ≤ ε . For any i = 1 , . . . , N , we have m ˙ Y ( { y i } ) = m Y ( B i ) ≤ lim inf n →∞ f n ∗ m X n ( B i ) = lim inf n →∞ m X n ( A n,i ) ≤ lim inf n →∞ Φ n ∗ m X n ( A i ) ≤ ν ( A i ) = g ∗ ν ( { y i } ) , ONVERGENCE OF METRIC TRANSFORMED SPACES 22 where the second inequality follows from Claim 4.10. Moreover, since m ˙ Y ( { y } ) ≤ ε , we have m ˙ Y ( B ) ≤ g ∗ ν ( B ) + ε for any subset B ⊂ ˙ Y , which implies that d P ( g ∗ ν, m ˙ Y ) ≤ ε .We next prove that the map g : F ( Z ) → ˙ Y is 1-Lipschitz up to 3 ε . Since ν ( Z \ e Z ) = g ∗ ν ( { y } ) ≤ m ˙ Y ( { y } ) ≤ ε, we see that ν ( e Z ) ≥ − ε . It is sufficient to prove that(4.2) d ˙ Y ( g ( z ) , g ( z ′ )) ≤ F ( k z − z ′ k ∞ ) + 3 ε holds for any z, z ′ ∈ e Z . We take any i, j ∈ { , . . . , N } and fix them. We take any points z ∈ A i and z ′ ∈ A j . There exist x n ∈ A n,i and x ′ n ∈ A n,j , n = 1 , , . . . , such that k Φ n ( x n ) − z k ∞ , k Φ n ( x ′ n ) − z ′ k ∞ → n → ∞ . Since we have d Y ( y i , y j ) ≤ d Y ( B i , B j ) + 2 ε ≤ d Y ( f n ( x n ) , f n (˜ x ′ )) + 2 ε ≤ F n ( d X n ( x n , ˜ x ′ )) + ε n + 2 ε for any ˜ x ′ ∈ A n,j , we obtain d Y ( y i , y j ) ≤ lim sup n →∞ F n ( d X n ( x n , A n,j )) + 2 ε. On the other hand, we see that d Y ( y i , y j ) ≤ max i,j =1 ,...,N d Y ( y i , y j ) ≤ F ( R ) + 3 ε = lim n →∞ F n ( R ) + 3 ε. Combining these implies that d Y ( y i , y j ) ≤ lim sup n →∞ F n (min { d X n ( x n , A n,j ) , R } ) + 3 ε = lim sup n →∞ F n ((Φ n ( x n )) j ) + 3 ε = lim sup n →∞ F n ((Φ n ( x n ) − Φ n ( x ′ n )) j ) + 3 ε. By (II) and Lemma 3.8, we have d Y ( y i , y j ) ≤ lim sup n →∞ F n ( k Φ n ( x n ) − Φ n ( x ′ n ) k ∞ ) + 3 ε. Furthermore, since k Φ n ( x n ) − Φ n ( x ′ n ) k ∞ → k z − z ′ k ∞ as n → ∞ , we have d ˙ Y ( g ( z ) , g ( z ′ )) = d Y ( y i , y j ) ≤ F ( k z − z ′ k ∞ ) + 3 ε by Lemma 3.7. Therefore we obtain (4.2). This completes the proof. (cid:3) Applying Corollary 4.7 to the map ι ◦ g : F ( Z ) → Y , we obtain Y ∈ F ( P ). This completesthe proof of Proposition 4.9. (cid:3) Proof of ‘(2) ⇒ (1)’ of Theorem 1.2. This follows from Proposition 4.8 and Proposition 4.9. (cid:3)
ONVERGENCE OF METRIC TRANSFORMED SPACES 23
Proof of Corollary 1.4 (A) . ‘(A1) ⇒ (A2)’ follows from Proposition 4.2 and Proposition 4.4.We prove ‘(A2) ⇒ (A1)’. Assume that P n converges weakly to P .We take any mm-space Y ∈ F ( P ) and any real number ε >
0. There exist two mm-spaces X ∈ P and Y ′ ∈ X such that F ( X ) dominates Y ′ and (cid:3) ( Y, Y ′ ) ≤ ε . Since P n convergesweakly to P , there exist mm-spaces X n ∈ P n , n = 1 , , . . . , such that (cid:3) ( X n , X ) → n → ∞ . Since, in particular, X n converges weakly to P X and Y ′ ∈ P F ( X ), we havelim n →∞ (cid:3) ( Y ′ , P F n ( X n )) = 0by Proposition 4.8. Thus we have (cid:3) ( Y, F n ( P n )) ≤ (cid:3) ( Y ′ , F n ( P n )) + ε ≤ (cid:3) ( Y ′ , P F n ( X n )) + ε, which implies that lim n →∞ (cid:3) ( Y, F n ( P n )) = 0.On the other hand, we assume that an mm-space Y satisfieslim inf n →∞ (cid:3) ( Y, F n ( P n )) = 0 . Then, there exist mm-spaces X n ∈ P n , n = 1 , , . . . , such thatlim inf n →∞ (cid:3) ( Y, P F n ( X n )) = 0 . By the compactness of Π, { X n } n ∈ N has a weak convergent subsequence. Let Q be a limitpyramid of a weak convergent subsequence of { X n } n ∈ N . By Proposition 4.9, we have Y ∈ F ( Q ). Moreover, since X n ∈ P n for each n , we have Q ⊂ P , which implies that F ( Q ) ⊂ F ( P ). Thus we obtain Y ∈ F ( P ).Combining these means that F n ( P n ) converges weakly to F ( P ). This completes the proofof this corollary. (cid:3) Proof of Theorem 1.3.
We start with proving the following lemma.
Lemma 4.12.
Let F n , F : [0 , + ∞ ) → [0 , + ∞ ) , n = 1 , , . . . , be continuous metric preservingfunctions satisfying (I) , (II) and (IV) . If lim n →∞ (cid:3) ( F ( Y ) , P F n ( X n )) = 0 for mm-spaces Y and X n , n = 1 , , . . . , then we have lim n →∞ (cid:3) ( Y, P X n ) = 0 . Proof.
Assume that (cid:3) ( F ( Y ) , P F n ( X n )) → n → ∞ . By Lemma 4.6, there exist Borelmeasurable maps f n : F n ( X n ) → F ( Y ) and a sequence ε n → f n is 1-Lipschitz upto ε n and fulfills d F ◦ d Y P ( f n ∗ m X n , m Y ) ≤ ε n for every n .We take any real number ε >
0. We prove the following: there exist an mm-space Z anda map h : Z → Y (both depending on ε ) such that • (cid:3) ( Z, P X n ) → n → ∞ , • h is 1-Lipschitz up to 3 ε and d P ( h ∗ m Z , m Y ) ≤ ε holds.If we prove this, then, by Lemma 4.6, we havelim sup n →∞ (cid:3) ( Y, P X n ) ≤ ε. Note that the outline of the following proof is similar to the proof of Proposition 4.9.
ONVERGENCE OF METRIC TRANSFORMED SPACES 24
We find finite many open sets B , . . . , B N ⊂ Y such that diam B i < ε and m Y ( B i ) > i ∈ { , . . . , N } and that δ := min ≤ i
For any i = 1 , . . . , N and for any sufficiently large n , we have Φ n ( A n,i ) ⊂ A i . Proof.
Take any i ∈ { , . . . , N } and fix it. It is sufficient to prove thatlim inf n →∞ d X n ( A n,i , A n,j ) ≥ δ for every j = i . We put α := lim inf n →∞ d X n ( A n,i , A n,j ). There exists a subsequence { n k } ⊂{ n } such that d X nk ( A n k ,i , A n k ,j ) → α as k → ∞ . We find two sequences { x m } m ⊂ A n k ,i and { x ′ m } m ⊂ A n k ,j such that d X nk ( x m , x ′ m ) → d X nk ( A n k ,i , A n k ,j ) as m → ∞ . Then we have F ( δ ) ≤ F ( d Y ( B i , B j )) ≤ F ( d Y ( f n ( x m ) , f n ( x ′ m ))) ≤ F n k ( d X nk ( x m , x ′ m )) + ε n k → F n k ( d X nk ( A n k ,i , A n k ,j )) + ε n k as m → ∞ ONVERGENCE OF METRIC TRANSFORMED SPACES 25 for each k . Moreover, by Lemma 3.7, we havelim k →∞ F n k ( d X nk ( A n k ,i , A n k ,j )) = F ( α ) . Combining these implies F ( δ ) ≤ F ( α ), so that δ ≤ α since F is increasing. This completesthe proof. (cid:3) We define a subset e Z ⊂ Z by e Z := supp ν ∩ N G i =1 A i and define a map g : Z → ˙ Y by g ( z ) := (cid:26) y i if z ∈ A i ,y if z ∈ Z \ e Z. If B = ∅ , then we set g ( z ), for z ∈ Z \ e Z , an arbitrary point of ˙ Y . Claim 4.14.
The map g : Z → ˙ Y is -Lipschitz up to ε and fulfills d P ( g ∗ ν, m ˙ Y ) ≤ ε. Proof.
The proof of d P ( g ∗ ν, m ˙ Y ) ≤ ε is completely same as in the proof of Claim 4.11. Weprove that the map g : Z → ˙ Y is 1-Lipschitz up to 2 ε . Since ν ( Z \ ˜ Z ) = g ∗ ν ( { y } ) ≤ m ˙ Y ( { y } ) ≤ ε, we see that ν ( ˜ Z ) ≥ − ε . It is sufficient to prove that(4.3) d ˙ Y ( g ( z ) , g ( z ′ )) ≤ k z − z ′ k ∞ + 2 ε holds for any z, z ′ ∈ ˜ Z . We take any i, j ∈ { , . . . , N } and fix them. We take any points z ∈ A i and z ′ ∈ A j . There exist x n ∈ A n,i and x ′ n ∈ A n,j , n = 1 , , . . . , such that k Φ n ( x n ) − z k ∞ , k Φ n ( x ′ n ) − z ′ k ∞ → n → ∞ . Since we have F ( d Y ( y i , y j ) − ε ) ≤ F ( d Y ( B i , B j )) ≤ F ( d Y ( f n ( x n ) , f n (˜ x ′ ))) ≤ F n ( d X n ( x n , ˜ x ′ )) + ε n for any ˜ x ′ ∈ A n,j , we obtain F ( d Y ( y i , y j ) − ε ) ≤ lim sup n →∞ F n ( d X n ( x n , A n,j )) . On the other hand, we see that F ( d Y ( y i , y j ) − ε ) ≤ F ( R ) = lim n →∞ F n ( R ) . Combining these implies that F ( d Y ( y i , y j ) − ε ) ≤ lim sup n →∞ F n (min { d X n ( x n , A n,j ) , R } ) + 3 ε = lim sup n →∞ F n ((Φ n ( x n )) j )= lim sup n →∞ F n ((Φ n ( x n ) − Φ n ( x ′ n )) j ) . ONVERGENCE OF METRIC TRANSFORMED SPACES 26
By (II) and Lemma 3.8, we have F ( d Y ( y i , y j ) − ε ) ≤ lim sup n →∞ F n ( k Φ n ( x n ) − Φ n ( x ′ n ) k ∞ ) . Furthermore, since k Φ n ( x n ) − Φ n ( x ′ n ) k ∞ → k z − z ′ k ∞ as n → ∞ , we have F ( d Y ( y i , y j ) − ε ) ≤ F ( k z − z ′ k ∞ ) , which leads to d ˙ Y ( g ( z ) , g ( z ′ )) = d Y ( y i , y j ) ≤ k z − z ′ k ∞ + 2 ε since F is increasing. Therefore we obtain (4.3). (cid:3) By Claim 4.14, the composition h := ι ◦ g : Z → Y is 1-Lipschitz up to 3 ε and d P ( h ∗ m Z , m Y ) ≤ ε holds. The proof is completed. (cid:3) Lemma 4.15.
Let F : [0 , + ∞ ) → [0 , + ∞ ) be a continuous nondecreasing metric preservingfunction. Then, the following (1) – (3) are equivalent to each other. (1) F is increasing. (2) If two pyramids P , Q ∈ Π satisfy F ( Q ) ⊂ F ( P ) , then Q ⊂ P . (3) The map e F : Π ∋ P 7→ F ( P ) ∈ Π is injective.Proof. ‘(2) ⇒ (3)’ is obvious.We first prove ‘(3) ⇒ (1)’. Suppose that F is not increasing. By F (0) = 0 and thecontinuity of F , there exist two positive real numbers s < s ′ such that F ( s ) = F ( s ′ ). Twomm-spaces X and Y are defined as X := ( { , s } , | · | , δ ,s ) and Y := ( { , s ′ } , | · | , δ ,s ′ ) , where the notation is same as in the proof of Proposition 4.2. Then, X and Y are notmm-isomorphic to each other, but F ( X ) = F ( Y ), so that e F is not injective.We next prove ‘(1) ⇒ (2)’. Assume that two pyramids P , Q ∈
Π satisfy F ( Q ) ⊂ F ( P ).We take any mm-space Y ∈ Q . Since F ( Y ) ∈ F ( Q ) ⊂ F ( P ), there exist mm-spaces X n ∈ P , n = 1 , , . . . , such that lim n →∞ (cid:3) ( F ( Y ) , P F ( X n )) = 0 . By Lemma 4.12, we obtain (cid:3) ( Y, P ) ≤ lim n →∞ (cid:3) ( Y, P X n ) = 0 , which implies Y ∈ P . This completes the proof of this lemma. (cid:3) Proposition 4.16.
Let F n , F : [0 , + ∞ ) → [0 , + ∞ ) be continuous metric preserving func-tions. Assume that F n converges pointwise to F as n → ∞ and that F is nondecreasing.Then, for any pyramid P ∈ Π , there exists a sequence { Y n } n ∈ N of mm-spaces such that Y n converges weakly to P and F n ( Y n ) converges weakly to F ( P ) .Proof. We take any pyramid
P ∈
Π. Let { Y m } m ∈ N be an approximation of P (see Lemma2.23). Note that Y m converges weakly to P as m → ∞ . For each m , by [4, Corollary 4.4](see Remark 1.6), we have lim n →∞ (cid:3) ( F n ( Y m ) , F ( Y m )) = 0 . ONVERGENCE OF METRIC TRANSFORMED SPACES 27
Choosing sufficiently small m than n , there exists a sequence { m ( n ) } n ∈ N with m ( n ) → ∞ as n → ∞ such that(4.4) lim n →∞ (cid:3) ( F n ( Y m ( n ) ) , F ( Y m ( n ) )) = 0 . We prove that F n ( Y m ( n ) ) converges weakly to F ( P ) as n → ∞ .Take any mm-space Z ∈ F ( P ) and any real number ε >
0. There exist mm-spaces X ∈ P and Z ′ ∈ X such that(4.5) Z ′ ≺ F ( X ) and (cid:3) ( Z, Z ′ ) < ε. Moreover, there exist mm-spaces X n , n = 1 , , . . . , such that Y m ( n ) dominates X n for all n and (cid:3) ( X n , X ) → n → ∞ . Since F is nondecreasing, we see that(4.6) F ( X n ) ≺ F ( Y m ( n ) ) for every n and lim n →∞ (cid:3) ( F ( X n ) , F ( X )) = 0 . Combining (4.4), (4.5), (4.6), and Lemma 2.11 means that, for any sufficiently large n , thereexist maps f n : F n ( Y m ( n ) ) → Z such that f n is 1-Lipschitz up to 4 ε and d P ( f n ∗ m Y m ( n ) , m Z ) ≤ ε . Thus, by Lemma 4.6, we have (cid:3) ( Z, P F n ( Y m ( n ) )) ≤ ε for any sufficiently large n . We obtain lim n →∞ (cid:3) ( Z, P F n ( Y m ( n ) )) = 0.On the other hand, we take any mm-space Z ∈ X such thatlim inf n →∞ (cid:3) ( Z, P F n ( Y m ( n ) )) = 0 . Taking a subsequence of n , we can assume that (cid:3) ( Z, P F n ( Y m ( n ) )) → n → ∞ . ByLemma 4.6, (4.4), and Lemma 2.11, there exist maps g n : F ( Y m ( n ) ) → Z and a sequence ε n → g n is 1-Lipschitz up to ε n and d P ( g n ∗ m Y m ( n ) , m Z ) ≤ ε n holds for every n .Since F ( Y m ( n ) ) ∈ F ( P ) follows from Y m ( n ) ∈ P , we have Z ∈ F ( P ) by Corollary 4.7.Therefore F n ( Y m ( n ) ) converges weakly to F ( P ) as n → ∞ . The proof is completed. (cid:3) Proof of ‘(1) ⇒ (2)’ of Theorem 1.3. Assume the condition (1). We first prove (I). By Propo-sition 3.11, it is sufficient to prove that if F n ( s n ) → F ( s ) for given positive real numbers s n , n = 1 , , . . . , and s , then we have s n → s . Let s n , n = 1 , , . . . , and s be positive real numberswith F n ( s n ) → F ( s ). We define mm-spaces X n and X as X n := ( { , s n } , | · | , δ ,s n ) and X := ( { , s } , | · | , δ ,s ) , where the notation is same as in the proof of Proposition 4.2. Note that F n ( X n ) (cid:3) -convergesto F ( X ). By the condition (1), X n concentrates to X . Thus we have s n → s as n → ∞ inthe same way as the proof of Proposition 4.2. We obtain (I). Moreover, (II) follows fromLemma 4.3 directly.We next prove (IV). By Lemma 4.15, it is sufficient to prove that the map P 7→ F ( P ) isinjective. We take any two pyramids P , Q and assume that F ( P ) = F ( Q ). By Proposition4.16, there exists a sequence { Y n } n ∈ N of mm-spaces such that Y n converges weakly to P and F n ( Y n ) converges weakly to F ( P ). It follows from F ( P ) = F ( Q ) and the condition (1) that Y n converges weakly to Q , which implies that P = Q . This completes the proof. (cid:3) ONVERGENCE OF METRIC TRANSFORMED SPACES 28
We next prove ‘(2) ⇒ (1)’ of Theorem 1.3. Let F n , F be continuous metric preservingfunctions satisfying (I) (II), and (IV). Take any sequence { X n } n ∈ N of mm-spaces and anypyramid P ∈
Π such that F n ( X n ) converges weakly to F ( P ). Our goal is to prove that X n converges weakly to P as n → ∞ . Proposition 4.17.
For any Y ∈ P , we have lim n →∞ (cid:3) ( Y, P X n ) = 0 . Proof.
We take any mm-space Y ∈ P . Since F ( Y ) ∈ F ( P ) and F n ( X n ) converges weakly to F ( P ), we see that lim n →∞ (cid:3) ( F ( Y ) , P F n ( X n )) = 0 . By Lemma 4.12, we have lim n →∞ (cid:3) ( Y, P X n ) = 0 . The proof is completed. (cid:3)
Proposition 4.18.
If an mm-space Y satisfies lim inf n →∞ (cid:3) ( Y, P X n ) = 0 , then Y ∈ P .Proof. Choosing a subsequence of n , we can assume that (cid:3) ( Y, P X n ) → n → ∞ . Then,by Lemma 4.6, there exist Borel measurable maps f n : X n → Y and a sequence ε n → f n is 1-Lipschitz up to ε n and d d Y P ( f n ∗ m X n , m Y ) ≤ ε n holds for every n . By Lemma 4.5,there exists a sequence δ n → f n : F n ( X n ) → F ( Y ) is 1-Lipschitz up to δ n and d F ◦ d Y P ( f n ∗ m X n , m Y ) ≤ δ n holds. By Lemma 4.6, we have (cid:3) ( F ( Y ) , P F n ( X n )) ≤ δ n → n → ∞ Since F n ( X n ) converges weakly to F ( P ), we have F ( Y ) ∈ F ( P ). Thus, since F is increasing,we obtain Y ∈ P . The proof is completed. (cid:3) Proof of ‘(2) ⇒ (1)’ of Theorem 1.3. This follows from Proposition 4.17 and Proposition 4.18. (cid:3)
Proof of Corollary 1.4 (B) . ‘(B1) ⇒ (B2)’ follows from Theorem 1.3 directly. We prove ‘(B2) ⇒ (B1)’. Assume that F n ( P n ) converges weakly to F ( P ).We take any mm-space Y ∈ P . Since F ( Y ) ∈ F ( P ), we havelim n →∞ (cid:3) ( F ( Y ) , F n ( P n )) = 0 . Then, there exist mm-spaces X n ∈ P n , n = 1 , , . . . , such thatlim n →∞ (cid:3) ( F ( Y ) , P F n ( X n )) = 0 . By Lemma 4.12, we have lim sup n →∞ (cid:3) ( Y, P n ) ≤ lim n →∞ (cid:3) ( Y, P X n ) = 0 . On the other hand, we assume that an mm-space Y satisfieslim inf n →∞ (cid:3) ( Y, P n ) = 0 . ONVERGENCE OF METRIC TRANSFORMED SPACES 29
Taking a subsequence of n , we can assume that there exist mm-spaces X n ∈ P n and Y n ∈ X , n = 1 , , . . . , such that X n dominates Y n for every n and (cid:3) ( Y n , Y ) → n → ∞ . Since all F n are nondecreasing, we see that F n ( X n ) dominates F n ( Y n ) for every n and (cid:3) ( F n ( Y n ) , F ( Y )) → n → ∞ . Thus, we havelim sup n →∞ (cid:3) ( F ( Y ) , F n ( P n )) ≤ lim n →∞ (cid:3) ( F ( Y ) , F n ( Y n )) = 0 , which implies F ( Y ) ∈ F ( P ), so that Y ∈ P since F is increasing.Combining these means that P n converges weakly to P . This completes the proof of thiscorollary. (cid:3) Box-convergence/concentration of metric transformed spaces
The goal in this section is to prove Theorem 1.5.
Proposition 5.1.
Let F n , F : [0 , + ∞ ) → [0 , + ∞ ) , n = 1 , , . . . , be continuous metric pre-serving functions. Assume that F n converges pointwise to F as n → ∞ . If there exists asequence s n → ∞ such that lim inf n →∞ F n ( s n ) < sup F, then there exist a sequence { X n } n ∈ N of mm-spaces such that F n ( X n ) (cid:3) -converges to F ( X ) for an mm-space X but X n does not concentrate.Proof. We take a sequence s n → ∞ such thatlim inf n →∞ F n ( s n ) < sup F. We set α := lim inf n →∞ F n ( s n ). There exists a real number β such that F ( β ) = α . We finda subsequence { n i } ⊂ { n } such thatlim i →∞ F n i ( s n i ) = α = F ( β ) . We define an mm-space X := ( { , β } , | · | , δ ,β ) , and define mm-spaces X n := (cid:26) ( { , s n i } , | · | , δ ,s ni ) if n = n i ,X otherwise,where the notation is same as in the proof of Proposition 4.2. It is easy to see that F n ( X n ) (cid:3) -converges to F ( X ). Moreover, since s n i → ∞ , we see that X n i converges weakly to thepyramid P := { ( { , s } , | · | , δ ,s ) | s ≥ } as i → ∞ . Since P is not (cid:3) -precompact, X n i does not concentrate to any mm-spaces. Thiscompletes the proof. (cid:3) The following proposition is a corollary of Lemma 4.15.
Proposition 5.2.
Let F : [0 , + ∞ ) → [0 , + ∞ ) be a continuous metric preserving function ( we do not assume that F is nondecreasing ) . Then, the following (1) and (2) are equivalentto each other. ONVERGENCE OF METRIC TRANSFORMED SPACES 30 (1) F is increasing. (2) The map
X ∋ X F ( X ) ∈ X is injective.Proof. The implication ‘(1) ⇒ (2)’ follows from Lemma 4.15 directly. The proof of ‘(2) ⇒ (1)’ is completely same as the proof of ‘(3) ⇒ (1)’ of Lemma 4.15. Note that we do not needthe assumption that F is nondecreasing in this proof. (cid:3) Proof of Theorem 1.5.
We first prove ‘(2) ⇒ (1)’ and ‘(3) ⇒ (1)’ together. Assume thatone of the conditions (2) or (3) holds. We have (I) by the completely same discussion asthe proof of Theorem 1.3. Moreover, (II) follows from Proposition 5.1 and Lemma 3.8. Weverify (IV). We take any two mm-spaces X and Y with F ( X ) = F ( Y ). By [4, Corollary 4.4](see also Remark 1.6), we see that F n ( X ) (cid:3) -converges to F ( X ). By F ( X ) = F ( Y ) and ourassumption, we have X = Y . Thus the map X ∋ X F ( X ) ∈ X is injective. Combiningthis and Proposition 5.2 implies that F is increasing. Thus we obtain ‘(2) ⇒ (1)’ and ‘(3) ⇒ (1)’.The implication ‘(1) ⇒ (3)’ follows from Theorem 1.3 directly. We next prove ‘(1) ⇒ (2)’.Assume that (I), (II) and (IV) hold. Take any sequence { X n } n ∈ N of mm-spaces and anymm-space X such that F n ( X n ) (cid:3) -converges to F ( X ). We prove that X n (cid:3) -converges to X .Take any real number ε >
0. By the inner regularity of m X , there exists a compact set K ⊂ X such that m X ( K ) ≥ − ε . Let D ε := (cid:26) F (diam K ) + 4 ε if sup F = + ∞ , ( F (diam K ) + sup F ) if sup F < + ∞ . Note that D ε < + ∞ and [0 , D ε ] ⊂ Im F . Since F is continuous and increasing, so is theinverse function F − : Im F → [0 , + ∞ ). Moreover, F − is uniformly continuous on [0 , D ε ].Let ω ε is the minimal modulus of continuity of F − | [0 ,D ε ] , that is, ω ε ( δ ) := sup (cid:8) | F − ( s ) − F − ( t ) | (cid:12)(cid:12) s, t ∈ [0 , D ε ] with | s − t | ≤ δ (cid:9) . We take any real number δ such that0 < δ < min (cid:26) ε, sup F − F (diam K )8 (cid:27) and ω ε ( δ ) < ε. Since F n ( X n ) (cid:3) -converges to F ( X ), there exist δ -mm-isomorphisms f n : F n ( X n ) → F ( X ) forsufficiently large n . Claim 5.3.
For every sufficiently large n , the map f n : X n → X is a ε -mm-isomorphism.Proof of Claim 5.3. We first prove d d X P ( f n ∗ m X n , m X ) ≤ ε . For any subset A ⊂ X , we have U F ◦ d X δ ( A ) ⊂ U d X F − ( δ ) ( A ) . In fact, taking any point y ∈ U F ◦ d X δ ( A ), it holds that F ( d X ( y, A )) < δ , which implies that d X ( y, A ) < F − ( δ ). Combining d F ◦ d X P ( f n ∗ m X n , m X ) ≤ δ and this leads to m X ( A ) ≤ f n ∗ m X n ( U F ◦ d X δ ( A )) + δ ≤ f n ∗ m X n ( U d X F − ( δ ) ( A )) + δ, which implies that d d X P ( f n ∗ m X n , m X ) ≤ max { F − ( δ ) , δ } . Since F − ( δ ) = | F − ( δ ) − F − (0) | ≤ ω ε ( δ ) < ε, ONVERGENCE OF METRIC TRANSFORMED SPACES 31 we obtain d d X P ( f n ∗ m X n , m X ) ≤ ε .Let X ′ n ⊂ F n ( X n ) be a nonexceptional domain of f n : F n ( X n ) → F ( X ) and let e X n := X ′ n ∩ f − n ( U F ◦ d X δ ( K )) . We see that m X n ( e X n ) ≥ m X n ( X ′ n ) + f n ∗ m X n ( U F ◦ d X δ ( K )) − ≥ m X ( K ) − δ ≥ − ε. It is sufficient to prove that we have | d X n ( x, x ′ ) − d X ( f n ( x ) , f n ( x ′ )) | ≤ ε for every sufficiently large n and for any x, x ′ ∈ e X n . We see that F ( d X ( f n ( x ) , f n ( x ′ ))) ≤ D ε , F n ( d X n ( x, x ′ )) ≤ D ε , and F ( d X n ( x, x ′ )) ≤ D ε for every sufficiently large n and for any x, x ′ ∈ e X n . In fact, we have F n ( d X n ( x, x ′ )) ≤ F ( d X ( f n ( x ) , f n ( x ′ ))) + δ ≤ F (diam K ) + 3 δ ≤ D ε . Suppose that d X n ( x, x ′ ) ≥ F − ( D ε + η ) for some η >
0. By Lemma 3.8, we have F n ( F − ( D ε + η )) ≤ F n ( d X n ( x, x ′ )) + δ ≤ D ε . As n → ∞ , this implies the contradiction D ε + η ≤ D ε . Thus we obtain d X n ( x, x ′ ) ≤ F − ( D ε ),that is, F ( d X n ( x, x ′ )) ≤ D ε . Combining this and Lemma 3.7 implies that, for any x, x ′ ∈ e X n , | d X n ( x, x ′ ) − d X ( f n ( x ) , f n ( x ′ )) |≤ | d X n ( x, x ′ ) − F − ( F n ( d X n ( x, x ′ ))) | + | F − ( F n ( d X n ( x, x ′ ))) − d X ( f n ( x ) , f n ( x ′ )) |≤ ω ε ( | F ( d X n ( x, x ′ )) − F n ( d X n ( x, x ′ )) | ) + ω ε ( | F n ( d X n ( x, x ′ )) − F ( d X ( f n ( x ) , f n ( x ′ ))) | ) ≤ ω ε ( δ ) < ε, where the third inequality follows from, for every sufficiently large n ,sup s ∈ [0 ,F − ( D ε )] | F ( s ) − F n ( s ) | ≤ δ. Thus the map f n : X n → X is a 3 ε -mm-isomorphism. The proof of this claim is completed. (cid:3) By Claim 5.3, we see that X n (cid:3) -converges to X as n → ∞ . Thus we obtain ‘(1) ⇒ (2)’.This completes the proof of Theorem 1.5. (cid:3) Application: spheres and projective spaces
Gaussian space.
Let λ be a positive real number. The product γ nλ := n O i =1 γ λ of the one-dimensional centered Gaussian measure γ λ of variance λ is the n -dimensionalcentered Gaussian measure on R n of variance λ . We call the mm-spaceΓ nλ := ( R n , k · k , γ nλ )the n -dimensional Gaussian space with variance λ . ONVERGENCE OF METRIC TRANSFORMED SPACES 32
For 1 ≤ k ≤ n , we denote by π nk : R n → R k the natural projection, that is, π nk ( x , x , . . . , x n ) := ( x , x , . . . , x k ) , ( x , x , . . . , x n ) ∈ R n . Since the projection π nn − : Γ nλ → Γ n − λ is 1-Lipschitz continuous and measure-preservingfor any n ≥
2, the Gaussian space Γ nλ is monotone nondecreasing in n with respect to theLipschitz order, so that, as n → ∞ , the Gaussian space Γ nλ converges weakly to the pyramid P Γ ∞ λ := ∞ [ n =1 P Γ nλ (cid:3) . We call P Γ ∞ λ the virtual Gaussian space with variance λ . We remark that the infiniteproduct measure γ ∞ λ := ∞ O i =1 γ λ is a Borel probability measure on R ∞ with respect to the product topology, but is not Borelwith respect to the l -norm.Let F = R , C , or H , where H is the algebra of quaternions, and let d := dim R F . Weconsider the Hopf action on Γ dnλ by identifying R dn with F n . Recall that the Hopf action isthe following U F (1)-action on F n : F n × U F (1) ∋ ( z, t ) zt ∈ F n , where U F (1) := { t ∈ F | k t k = 1 } is a group under multiplication. Since the projection π dndk : F n → F k , k ≤ n , is U F (1)-equivariant (i.e., π dndk ( zt ) = π dndk ( z ) t for any t ∈ U F (1) and for any z ∈ F n ), there exists a unique map ¯ π dndk : F n /U F (1) → F k /U F (1) such that q ◦ π dndk = ¯ π dndk ◦ q , where q is the quotient map of the Hopf action.The Hopf action is isometric with respect to the Euclidean distance and also preserves theGaussian measure γ dnλ . LetΓ dnλ /U F (1) := ( F n /U F (1) , d F n /U F (1) , ¯ γ dnλ )be the quotient space with the induced mm-structure, that is, d F n /U F (1) ([ z ] , [ w ]) := inf z ′ ∈ [ z ] ,w ′ ∈ [ w ] k z ′ − w ′ k , [ z ] , [ w ] ∈ F n /U F (1) , ¯ γ dnλ ( A ) := γ dnλ ( { z ∈ F n | [ z ] ∈ A } ) , A ⊂ F n /U F (1) . Since the map ¯ π dnd ( n − : Γ dnλ /U F (1) → Γ d ( n − λ /U F (1) is 1-Lipschitz continuous and measure-preserving, the quotient space Γ dnλ /U F (1) is monotone increasing in n with respect to theLipschitz order. The Hopf quotient space Γ dnλ /U F (1) converges weakly to the pyramid P Γ ∞ λ /U F (1) := ∞ [ n =1 P Γ dnλ /U F (1) (cid:3) . ONVERGENCE OF METRIC TRANSFORMED SPACES 33
Weak convergence of spheres and projective spaces.
Let S n ( r ) be the n -dimensionalsphere in R n +1 centered at the origin and of radius r >
0. We equip S n ( r ) with the standardRiemannian distance function d S n ( r ) or the restriction of the Euclidean distance function k · k .We also equip S n ( r ) with the Riemannian volume measure σ n normalized as σ n ( S n ( r )) = 1.Then S n ( r ) is an mm-space. We consider the Hopf quotient F P n ( r ) := S d ( n +1) − ( r ) /U F (1)that has a natural mm-structure induced from that of S d ( n +1) − ( r ) by the same way as above.This is topologically an n -dimensional projective space over F . Note that, if F = C and ifthe distance function on S n +1 ( r ) is assumed to be Riemannian, then the distance functionon C P n ( r ) coincides with that induced from the Fubini-Study metric scaled with factor r . Theorem 6.1 ([14, Theorem 8.1.1], [15, Corollary 1.3]) . Let { r n } ∞ n =1 be a given sequenceof positive real numbers, and let λ n := r n / √ n ( resp. λ n := r n / √ dn ) . Then we have thefollowing (1) and (2) . (1) { S n ( r n ) } n ∈ N ( resp. { F P n ( r n ) } n ∈ N ) is L´evy family ( i.e., concentrating to a one pointspace ) if and only if λ n converges to as n → ∞ . (2) { S n ( r n ) } n ∈ N ( resp. { F P n ( r n ) } n ∈ N ) infinitely dissipates if and only if λ n diverges toinfinity as n → ∞ . We omit to state the definition of the infinite dissipation. Dissipation is the opposite notionto concentration. The above theorem claims that the critical scale order for concentrationis √ n . Moreover, in the Euclidean case, the limit of spheres and projective spaces with thecritical scale order is known. Theorem 6.2 ([14, Theorem 8.1.1], [15, Theorem 1.2]) . Let { r n } ∞ n =1 be a given sequence ofpositive real numbers, and let λ n := r n / √ n ( resp. λ n := r n / √ dn ) . Assume that S n ( r n ) and F P n ( r n ) have the Euclidean distance function. As n → ∞ , S n ( r n ) ( resp. F P n ( r n )) convergesweakly to P Γ ∞ λ ( resp. P Γ ∞ λ /U F (1)) if and only if λ n converges to a positive real number λ .Remark . The ‘only if’ part of the above theorem can be easily checked as follows. In[10], the κ -observable diameter of a pyramid, which is a fundamental invariant of a pyra-mid, is introduced and the limit formula is proved. The κ -observable diameter of P Γ ∞ λ (resp. P Γ ∞ λ /U F (1)) is proportional to λ , so that the pyramid P Γ ∞ λ (resp. P Γ ∞ λ /U F (1)) isdifferent for each λ .Roughly speaking, this theorem has been obtained by proving • the limit of S n ( r n ) (resp. F P n ( r n )) dominates P Γ ∞ λ (resp. P Γ ∞ λ /U F (1)), • P Γ ∞ λ (resp. P Γ ∞ λ /U F (1)) dominates the limit of S n ( r n ) (resp. F P n ( r n ))from the construction of maps for the domination. We remark that it would be difficult tofind a map for the domination directly in the Riemannian case. Our goal is to prove Theorem1.7 using the convergence of metric transformed spaces. Proof of Theorem 1.7.
Let { r n } ∞ n =1 be a given sequence of positive real numbers. We definemetric preserving functions F n : [0 , + ∞ ) → [0 , + ∞ ), n = 1 , , . . . , by F n ( s ) := (cid:26) r n sin s r n if s ≤ πr n , r n if s > πr n . ONVERGENCE OF METRIC TRANSFORMED SPACES 34
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Department of Mathematics, Osaka University, Toyonaka, Osaka 560-0032, Japan
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