Glivenko-Cantelli Theory, Ornstein-Weiss quasi-tilings, and uniform Ergodic Theorems for distribution-valued fields over amenable groups
GGLIVENKO–CANTELLI THEORY, ORNSTEIN–WEISS QUASI-TILINGS,AND UNIFORM ERGODIC THEOREMS FOR DISTRIBUTION-VALUEDFIELDS OVER AMENABLE GROUPS
CHRISTOPH SCHUMACHER, FABIAN SCHWARZENBERGER, AND IVAN VESELIĆ
Abstract.
We consider random fields indexed by finite subsets of an amenable discretegroup, taking values in the Banach-space of bounded right-continuous functions. The field isassumed to be equivariant, local, coordinate-wise monotone, and almost additive, with finiterange dependence. Using the theory of quasi-tilings we prove an uniform ergodic theorem,more precisely, that averages along a Foelner sequence converge uniformly to a limitingfunction. Moreover we give explicit error estimates for the approximation in the sup norm. Introduction
Ergodic theorems for Banach space valued functions or fields have been studied among othersin [6, 7, 11] in a combinatorial setting. The three quoted papers consider different groupactions in increasing generality: the lattice Z d , monotilable amenable discrete groups andgeneral amenable discrete groups, respectively. Note that amenability is a natural assumptionfor the validity of the ergodic theorem, as shown explicitly in [14]. Already before thatcombinatorial ergodic theorems for Banach space valued functions have been proven in thecontext of Delone dynamical systems, see [8] and the references therein.The combinatorial framework offers the advantage of a minimum of probabilistic or measuretheoretic assumptions, the necessary one being that frequencies or densities of finite patternsare well defined and can be approximated by an exhaustion (corresponding to a law of largenumbers). A disadvantage of the combinatorial approach chosen, is that the range of colours(or the alphabet corresponding to the values of the random variables) needs to be finite. Also,the derived ergodic theorems are in a sense conditional: The convergence bound depends onthe speed of convergence of the pattern frequencies.Our present research aims at dispensing with the finiteness condition on the set of colours.The price to pay is that we have to assume more probabilistic structure and in particularindependence or at least finite range correlations. In return, this structure yields automaticallyquantitative approximation error bounds. No extra assumptions on the speed of convergence ofthe pattern frequencies is needed. For the case of fields defined over Z d and Z d -actions we haveestablished such an ergodic theorem in [12], which takes on the form of a Glivenko–Cantellitheorem, and which we recall now in an informal way. MSC: 60F99, 60B12, 62E20, 60K35.Keywords: Føler sequence, amenable group, quasi-tilings Glivenko–Cantelli theory, Uniform convergence,Empirical measures.November 11, 2018, 2017-11-09-SSV-amenable.tex. a r X i v : . [ m a t h - ph ] N ov CHRISTOPH SCHUMACHER, FABIAN SCHWARZENBERGER, AND IVAN VESELIĆ
Theorem A ([12]) . Let Λ n “ r , n q d X Z d , and ω “ p ω g q g P Z d P R Z d be an i. i. d. sequence ofreal random variables. Assume the field f : P p Z d q ˆ R Z d Ñ B : “ t D : R Ñ R | D right-continuous and bounded u is Z d -equivariant, monotone in each coordinate ω g , local, and almost additive, i. e. for disjoint Q , . . . , Q n Ď Z d and Q : “ Ť ni “ Q i we have (cid:13)(cid:13)(cid:13) f p Q, ω q ´ n ÿ i “ f p Q i , ω q (cid:13)(cid:13)(cid:13) ď n ÿ i “ (cid:12)(cid:12) B Q i (cid:12)(cid:12) ,where B Q i denotes the boundary set. Assume furthermore, that f : “ sup ω k f p id , ω q k ă 8 .Then there is a function f ˚ : R Ñ R such that for each m P N , there exist a p m q , b p m q ą ,such that for all j P N , j ą m , there is an event Ω j,m Ď R Z d , with the properties P p Ω j,m q ě ´ b p m q exp ` ´ a p m q | Λ j | ˘ and @ ω P Ω j,m : (cid:13)(cid:13)(cid:13)(cid:13) f p Λ j , ω q | Λ j | ´ f ˚ (cid:13)(cid:13)(cid:13)(cid:13) ď d ` ´ p d ` ` f q m d ` j ´ m ` m ¯ In particular, almost surely we have lim n Ñ8 (cid:13)(cid:13) f p Λ n , ‚q | Λ n | ´ f ˚ (cid:13)(cid:13) “ . For a precise formulation of the properties of the field f see Section 2. Let us note that in ourTheorem f takes values in the Banach space B of right continuous and bounded functionswith sup-norm while in [6, 7, 11] an arbitrary Banach space was allowed. This restriction isdue to our use of the Glivenko–Cantelli theory in the proof and currently we do not knowhow to extend it to arbitrary Banach spaces.Naturally one asks whether the above result and its proof extend to general finitely generatedamenable groups. In this case, obviously, the boundary has to be taken with respect to agenerating set S Ď G , and the sequence of squares Λ n has to be replaced by a Følner sequence.Indeed, if G satisfies additionally( ‘ ) There exists a Følner sequence p Λ n q n P N in G , and a sequence of symmetric grids T n “ T ´ n Ď G such that G “ Ť t P T n Λ n t is a disjoint union.the proofs of [12] apply with technical, but no strategic, modifications, as sketched in AppendixB.However, it is not clear in which generality assumption ( ‘ ) holds. In fact, the existenceof tiling Følner sequences (for general amenable groups) has been investigated in severalinstances. It turned out that there exist useful additional conditions which imply the validityof ( ‘ ), cf. [16, 5]. For instance, a group which is residually finite and amenable contains atiling Følner sequence. Unfortunately, there is a lack of the complete picture: It is still anopen question whether there exists a tiling Følner sequence in each amenable group.Since this question seems hard to answer, Ornstein and Weiss invented in [10] the theory of ε -quasi tilings. The idea is to consider a tiling which is in several senses weaker as the one in( ‘ ). For a given ε ą ‚ the group is not tiled with one element of a Følner sequence, but with finitely manyelements of this sequence; the number of these elements depends on ε ; LIVENKO–CANTELLI THEORY FOR ALMOST ADDITIVE FIELDS ON AMENABLE GROUPS 3 ‚ the tiles are allowed to overlap, but the proportion of the part of any tile which isallowed to intersect other tiles is at most of size ε . This property is called ε -disjointness; ‚ each element of a Følner sequence with a sufficiently large index is, up to a proportionof size ε the union of ε -disjoint tiles.The authors showed that each amenable group can be ε -quasi tiled. In [11] these ideas havebeen developed further in order to obtain quantitative estimates on the portion which iscovered by translates of one specific element of the tiles. The proof of our main result, whichwe state now in an informal way, is based on these results on quasi tilings. Theorem B.
Let p Λ n q be a Følner sequence in a finitely generated group G . Let ω “ p ω g q g P G P R G be an i. i. d. sequence of real random variables. Assume the field f : P p G q ˆ R G Ñ t D : R Ñ R | D right-continuous and bounded u , is G -equivariant, monotone in each coordinate ω g , local, and almost additive, i. e. for disjoint Q , . . . , Q n Ď G and Q : “ Ť ni “ Q i we have (cid:13)(cid:13)(cid:13) f p Q, ω q ´ n ÿ i “ f p Q i , ω q (cid:13)(cid:13)(cid:13) ď n ÿ i “ (cid:12)(cid:12) B Q i (cid:12)(cid:12) ,where B Q i denotes the boundary relative to a set of generators S Ď G . Assume furthermore,that f : “ sup ω k f p id , ω q k ă 8 .Then there is a function f ˚ : R Ñ R such that for each δ P p , q , there exist a p δ q ą , suchthat for all sufficiently large j P N , there is an event Ω j,δ Ď R G , with the properties P p Ω j,δ q ě ´ exp ` ´ a p δ q | Λ j | ˘ and @ ω P Ω j,δ (cid:13)(cid:13)(cid:13)(cid:13) f p Λ j , ω q | Λ j | ´ f ˚ (cid:13)(cid:13)(cid:13)(cid:13) ď p f ` | S | ` q δ .In particluar, almost surely we have lim n Ñ8 (cid:13)(cid:13)(cid:13) f p Λ n , ‚q | Λ n | ´ f ˚ (cid:13)(cid:13)(cid:13) “ . For a precise formulation, see Definition 2.2 and Theorem 2.5. To achieve the error bound inthe theorem, we work with an ε -quasi tiling with ε “ δ . Remark . Let us sketch the difference between the proof of Theorem B (see also Theorem 2.5below) and the Theorem 2.8 of [12] sketched as Theorem A above. There we heavily reliedon the fact that Z d can be tiled exactly with any cube of integer length. Since a generaldiscrete amenable group need not have such a tiling, we have to modify the geometric parts of the proof and use ε -quasi tilings as in [10, 11]. Since quasi tilings in general overlap, weloose independence of the corresponding random variables. This requires a change in the probabilistic part of the proof and in particular the use of resampling.The structure of the paper is as follows. In Section 2, we precisely describe the model andour result. In Section 3 we summarize results about ε -quasi tilings, which are fundamentalfor our proof. The error estimate in the main theorem and the corresponding approximationprocedure naturally split in three parts, which are treated consecutively in Sections 4 to 6.Section 4 is of geometric nature. Section 5 is based on multivariate Glivenko–Cantelli theory.Section 6 is geometric in spirit again. In the Appendix we prove a resampling lemma andindicate how the proof of [12] could be adapted to cover monotileable amenable groups. CHRISTOPH SCHUMACHER, FABIAN SCHWARZENBERGER, AND IVAN VESELIĆ Model and main results
We start this section with the introduction of the geometric and probabilistic setting: Werecall the notion of a Cayley graph of an amenable group G , introduce random colorings ofvertices, and define so-called admissible fields, which are random functions mapping finitesubsets of G to functions on R and satisfying a number of natural properties, cf. Definition 2.2.We are then in the position to formulate our main Theorem 2.5.Let G be a finitely generated group and S “ S ´ Ď G zt id u a finite generating system.Obviously G is countable. The set of all finite subsets of G is denoted by F and is countableas well. Throughout this paper we will assume that G is amenable, i. e. there exists a squence p Λ n q n P N of elements in F such that for each K P F one has(2.1) | Λ n K Λ n || Λ n | n Ñ8 ÝÝÝÑ K Λ n : “ t kg | k P K, g P Λ n u is the pointwise group multiplication of sets, Λ n K Λ n denotes the symmetric difference between the sets Λ n and K Λ n , and | A | denotes the cardinalityof the finite set A . A sequence p Λ n q n P N satisfying property (2.1) is called Følner sequence .The pair p G, S q gives rise to an undirected graph Γ p G, S q “ p
V, E q with vertex set V : “ G and edge set E : “ tt x, y u | xy ´ P S u . The graph Γ p G, S q is known as the Cayley graph of G with respect to the generating system S . Note that by symmetry of S the edge set E is well-defined. Let d : G ˆ G Ñ N denote the usual graph metric of Γ p G, S q . The distancebetween two non-empty sets Λ , Λ Ď G is given by d p Λ , Λ q : “ min t d p x, y q | x P Λ , y P Λ u .In the case where Λ “ t x u consists of only one element, we write d p x, Λ q for d pt x u , Λ q . Thediameter of a non-empty set Λ P F is defined by diam p Λ q : “ max t d p x, y q | x, y P Λ u .Given r ě
0, the r -boundary of a set Λ Ď G is defined by B r p Λ q : “ t x P Λ | d p x, G z Λ q ď r u Y t x P G z Λ | d p x, Λ q ď r u .and besides this we use the notationΛ r : “ Λ zB r p Λ q “ t x P Λ | d p x, G z Λ q ą r u .It is easy to verify that for a given Følner sequence p Λ n q n P N , or p Λ n q for short, and r ě n Ñ8 | B r p Λ n q || Λ n | “ n Ñ8 | Λ rn || Λ n | “ p Λ n q is a Følner sequence, then for arbitrary r ě p Λ rn q is a Følnersequence as well. Conversely, in order to show that a given sequence p Λ n q is a Følner sequence,it is sufficient [1, 13] to show for n Ñ 8 either | Λ n S Λ n || Λ n | Ñ | B p Λ n q || Λ n | Ñ G (or equivalently colorings of the vertices of Γ p G, S q ).We choose a (finite or infinite) set of possible colors A P B p R q . The sample set,Ω “ A G “ t ω “ p ω g q g P G | ω j P A u , LIVENKO–CANTELLI THEORY FOR ALMOST ADDITIVE FIELDS ON AMENABLE GROUPS 5 is the set of all possible colorings of G . Note that G acts in a natural way via translationson Ω. To be precise, we define for each g P G (2.4) τ g : Ω Ñ Ω, p τ g ω q x “ ω xg , p x P G q .Next, we introduce random colorings. As the σ -algebra we choose B p Ω q , the product σ -algebraon Ω generated by cylinder sets. Oftentimes, we are interested in (finite) products of A embedded in the infinite product space Ω. To this end, we set for Λ Ď G Ω Λ : “ A Λ : “ tp ω g q g P Λ | ω g P A u and define Π Λ : Ω Ñ Ω Λ by p Π Λ p ω qq g : “ ω g for each g P Λ . As shorthand notation we write ω Λ instead of Π Λ p ω q . Having introduced the measurablespace p Ω , B p Ω qq , we choose a probability measure P with the following properties:(M1) equivariance: For each g P G we have P ˝ τ ´ g “ P .(M2) existence of densities: There is a σ -finite measure µ on p A , B p A qq , such that foreach Λ P F the measure P Λ : “ P ˝ Π ´ is absolutely continuous with respect to µ Λ : “ Â g P Λ µ on Ω Λ . We denote the corresponding probability density functionby ρ Λ .(M3) independence condition: There exists r ě n P N and non-emptyΛ , . . . , Λ n P F with min t d p Λ i , Λ j q | i ‰ j u ą r we have ρ Λ “ ś nj “ ρ Λ j , whereΛ “ Ť nj “ Λ j .The measure P Λ is called the marginal measure of P . It is defined on p Ω Λ , B p Ω Λ qq , whereagain B p Ω Λ q is generated by the corresponding cylinder sets. Remark . (a) The constant r ě r “ P is a product measure.In the following, we consider partial orderings on Ω and on R k , respectively. Here we write ω ď ω for ω, ω P Ω, if for all g P G we have ω g ď ω g . The notion x ď x for x, x P R k isdefined in the same way. We consider the Banach space B : “ t F : R Ñ R | F right-continuous and bounded u ,which is equipped with supremum norm k ¨ k : “ k ¨ k . Definition 2.2.
A field f : F ˆ Ω Ñ B is called admissible if the following conditions aresatisfied(A1) equivariance: for Λ P F , g P G and ω P Ω we have f p Λ g, ω q “ f p Λ , τ g ω q .(A2) locality: for all Λ P F and ω, ω P Ω satisfying Π Λ p ω q “ Π Λ p ω q we have f p Λ , ω q “ f p Λ , ω q . CHRISTOPH SCHUMACHER, FABIAN SCHWARZENBERGER, AND IVAN VESELIĆ (A3) almost additivity: for arbitrary ω P Ω, pairwise disjoint Λ , . . . , Λ n P F and Λ : “ Ť ni “ Λ i we have (cid:13)(cid:13)(cid:13) f p Λ , ω q ´ n ÿ i “ f p Λ i , ω q (cid:13)(cid:13)(cid:13) ď n ÿ i “ b p Λ i q ,where b : F Ñ r , satisfies ‚ b p Λ q “ b p Λ g q for arbitrary Λ P F and g P G , ‚ D D f ą b p Λ q ď D f | Λ | for arbitrary Λ P F , ‚ lim i Ñ8 b p Λ i q{ | Λ i | “
0, if p Λ i q i P N is a Følner sequence. ‚ for Λ , Λ P F we have b p Λ Y Λ q ď b p Λ q ` b p Λ q , b p Λ X Λ q ď b p Λ q ` b p Λ q , and b p Λ z Λ q ď b p Λ q ` b p Λ q .(A4) monotonicity: f is antitone with respect to the partial orderings on Ω Ď R G and B ,i. e. if ω, ω P Ω satisfy ω ď ω , we have f p Λ , ω qp x q ě f p Λ , ω qp x q for all x P R and Λ P F .(A5) boundedness: sup ω P Ω k f pt id u , ω q k ă 8 . Remark . ‚ Locality (A2) can be formulated as follows: f p Λ , ¨ q is σ p Π Λ q -measurable.This enables us to define f Λ : Ω Λ Ñ B by f Λ p ω Λ q : “ f p Λ , ω q with Λ P F and ω P Ω. ‚ We call the function b in (A3) boundary term . Note that the fourth assumption on b in (A3) was not made in [12]. Indeed, this inequality is used to separate overlappingtiles and is unnecessary as soon as the group has the tiling property ( ‘ ). This fourthpoint is used only in Lemmas 5.3 and 3.5. ‚ The antitonicity assumption in (A4) can be weakend. In particular, our proofs apply tofields which are monotone in each coordinate, where the direction of the monotonicitycan be different for distinct coordinates. For simplicity reasons and as our mainexample (see [12]) satisfies (A4), we restrict ourselves to this kind of monotonicity. ‚ As shown in [12], a combination of (A1), (A3) and (A5)implies that the bound K f : “ sup t k f p Λ , ω q k { | Λ | | ω P Ω , Λ P F u ď D f ` sup ω P Ω k f pt id u , ω q k ă 8 .(2.5) Definition 2.4.
A set U of admissible fields is called admissible set , if their bound is uniform: K U : “ sup f P U K f ă 8 and each for each f P U condition (A3) is satisfied with the same boundary term b . In thissituation we denote the constant in (A3) by D U .Let us state the main theorem of this paper. Theorem 2.5.
Let G be a finitely generated amenable group with a Følner sequence p Λ n q .Further, let A P B p R q and p Ω “ A G , B p Ω q , P q a probability space such that P satisfies (M1)to (M3). Finally, let U be an admissible set. LIVENKO–CANTELLI THEORY FOR ALMOST ADDITIVE FIELDS ON AMENABLE GROUPS 7 (a) Then, there exists an event ˜Ω P B p Ω q such that P p ˜Ω q “ and for any f P U there exists afunction f ˚ P B , which does not depend on the specific Følner seqeunce p Λ n q , with @ ω P ˜Ω : lim n Ñ8 (cid:13)(cid:13)(cid:13)(cid:13) f p Λ n , ω q | Λ n | ´ f ˚ (cid:13)(cid:13)(cid:13)(cid:13) “ .(b) Furthermore, for each ε P p , { q , there exist j p ε q P N , independent of K U , and a p ε, K U q , b p ε, K U q ą , such that for all j P N , j ě j p ε q , there is an event Ω j,ε,K U P B p Ω q ,with the properties P p Ω j,ε,K U q ě ´ b p ε, K U q exp ` ´ a p ε, K U q | Λ j | ˘ and (cid:13)(cid:13)(cid:13)(cid:13) f p Λ j , ω q | Λ j | ´ f ˚ (cid:13)(cid:13)(cid:13)(cid:13) ď p K U ` D U ` q? ε for all ω P Ω j,ε,K U and all f P U . For examples of measures P satisfying (M1) to (M3) and of admissible fields, we refer to [12].The generalization of the geometry from the lattice Z d to an amenable group G does notaffect the examples. See also [15, 9] for a discussion of models giving rise to a discontinuousintegrated density of states, which nevertheless can be uniformly approximated by almostadditive fields. 3. Outline of ε -quasi tilings Let us give a brief introduction to the theory of ε -quasi tilings. The main ideas go back toOrnstein and Weiss in [10]. However the specific results we use here are taken from [11], seealso [13].Let p Q n q be a Følner sequence. This sequence is called nested , if for all n P N we have t id u Ď Q n Ď Q n ` . Using tranlations and subsequences it is easy to show that every amenablegroup contains a nested Følner sequence, c. f. [11, Lemma 2.6].We will use the elements of the nested Følner sequence p Q n q to ε -quasi tile elements of a givenFølner sequence p Λ j q for (very) large index j . The next definition provides the notion of an α -covering, ε -disjointness, and ε -quasi tiling. Definition 3.1.
Let G be a finitely generated group, α, ε P p , q , and I some index set. ‚ The sets Q i P F , i P I , are said to α -cover the set Λ P F , if(i) Ť i P I Q i Ď Λ, and(ii) | Λ X Ť i P I Q i | ě α | Λ | . ‚ The sets Q i P F , i P I , are called ε -disjoint , if there are subsets ˚ Q i Ď Q i , i P I , suchthat for all i P I we have(i) | Q i z ˚ Q i | ď ε | Q i | , and(ii) ˚ Q i and Ť j P I zt i u ˚ Q j are disjoint. ‚ The K i P F , i P I , are said to ε -quasi tile Λ P F , if there exist T i P F , i P I , such that(i) the elements of t K i T i | i P I u are pairwise disjoint,(ii) for each i P I , the elements of t K i t | t P T i u are ε -disjoint, and(iii) the family t K i T i | i P I u p ´ ε q -covers Λ. CHRISTOPH SCHUMACHER, FABIAN SCHWARZENBERGER, AND IVAN VESELIĆ
The set T i is called center set for the tile K i , i P I .Actually, the details in this definition are adapted to our needs in this paper, as is the followingtheorem. The general and more technical versions as well as the proof of can be found [11].See also [10] for earlier results.Roughly speaking, the following theorem provides, in the setting of finitely generated amenablegroups, ε -quasi covers for every set with small enough boundary compared to its volume.Additionally, the theorem also provides control on the fraction covered by different tileswith uniform almost densities. To quantify these densities, we use the standard notation r b s : “ inf t z P Z | z ě b u “ inf Z X r b, for the smallest integer above b P R and define, forall ε ą i P N ,(3.1) N p ε q : “ R ln p ε q ln p ´ ε q V and η i p ε q : “ ε p ´ ε q N p ε q´ i . Theorem 3.2.
Let G be a finitely generated amenable group, p Q n q a nested Følner sequence,and ε P p , { q . Then there is a finite and strictly increasing selection of sets K i P t Q n | n P N u , i P t , . . . , N p ε qu , with the following quasi tiling property. For each Følner sequence p Λ j q ,there exists j p ε q P N such that for all j ě j p ε q , the sets K i , i P t , . . . , N p ε qu , are an ε -quasitiling of Λ j . Moreover, for all j ě j p ε q and all i P t , . . . , N p ε qu , the proportion of Λ j coveredby the tile K i satisfies (3.2) (cid:12)(cid:12)(cid:12)(cid:12) | K i T ji || Λ j | ´ η i p ε q (cid:12)(cid:12)(cid:12)(cid:12) ď ε N p ε q ,where T ji denotes the center set of the tile K i for the ε -quasi cover of Λ j . To make full use of Theorem 3.2, we need some properties of the densities η i p ε q . Lemma 3.3.
For N p ε q and η i p ε q as in (3.1) , the following holds true.(a) For each ε P p , q we have ´ ε ď N p ε q ÿ i “ η i p ε q “ ´ p ´ ε q N p ε q ď .(b) For each ε P p , { q and i P t , . . . , N p ε qu , we have εN p ε q ď η i p ε q ď ε .(c) For a bounded sequence p α i q i P N and ε P p , { q we have the inequality (cid:12)(cid:12)(cid:12)(cid:12) N p ε q ÿ i “ α i η i p ε q (cid:12)(cid:12)(cid:12)(cid:12) ď A ? ε ` A ε , where A : “ sup t | α i | | i P N u and A ε : “ sup t | α i | | i P N , i ě ε ´ { u . In particular, lim ε Œ N p ε q ÿ i “ α i η i p ε q ď lim inf i Ñ8 | α i | . LIVENKO–CANTELLI THEORY FOR ALMOST ADDITIVE FIELDS ON AMENABLE GROUPS 9
Proof.
Part (a) is an easy implication of the sum formula for the geometric series. We referto [11, Remark 4.3] for the details.Let us prove (b). By definition of η i p ε q we have η i p ε q ď ε . In order to see the other inequality,we note that η i p ε q ě ε p ´ ε q N p ε q´ ě ε p ´ ε q ln p ε q ln p ´ ε q “ ε .Thus, it is sufficient to show that ε ě { N p ε q . To this end, note that by definition of N p ε q the following holds true: εN p ε q ě ε ln p ε q ln p ´ ε q .Using the assumption ε P p , { q , a short and elementary calculation shows that the lastexpression is bounded from below by 1.To verify part (c), set N ˚ ε : “ t ε ´ { u : “ sup Z X p´8 , ε ´ { s , and calculate as follows (cid:12)(cid:12)(cid:12)(cid:12) N p ε q ÿ i “ α i η i p ε q (cid:12)(cid:12)(cid:12)(cid:12) ď (cid:12)(cid:12)(cid:12)(cid:12) N ˚ ε ÿ i “ α i η i p ε q (cid:12)(cid:12)(cid:12)(cid:12) ` (cid:12)(cid:12)(cid:12)(cid:12) N p ε q ÿ i “ N ˚ ε ` α i η i p ε q (cid:12)(cid:12)(cid:12)(cid:12) ď AN ˚ ε ε ` A ε ď A ? ε ` A ε .Note that it is easy to show that for 0 ă ε ă {
10 we have N p ε q ą N ˚ ε ą
0, such that bothsums are non-empty. (cid:44)
Next, we derive a useful corollary of Theorem 3.2.
Corollary 3.4.
Let a finitely generated group G , a subset Λ P F and ε P p , { q be given.Assume furthermore that the sets K i P F , i P t , . . . , N p ε qu , are an ε -quasi tiling of Λ withalmost densities η i p ε q and center sets T i P F , i P t , . . . , N p ε qu , satisfying (3.2) . Then wehave for each i P t , . . . , N p ε qu , the inequality estimating the “density” of the tile K i : (cid:12)(cid:12)(cid:12)(cid:12) | T i || Λ | ´ η i p ε q | K i | (cid:12)(cid:12)(cid:12)(cid:12) ď ε η i p ε q | K i | .Proof. We fix i P t , . . . , N p ε qu , employ ε -disjointness and the density estimate (3.2), anddeduce p ´ ε q | K i || T i || Λ | ď | K i T i || Λ | ď η i p ε q ` ε N p ε q .Therefore, with part (b) of Lemma 3.3, we get | T i || Λ | ´ η i p ε q | K i | ď η i p ε q ` ε N p ε q p ´ ε q | K i | ´ η i p ε q | K i | “ εη i p ε q ` ε N p ε q p ´ ε q | K i | ď εη i p ε qp ´ ε q | K i | ď εη i p ε q | K i | .Equation (3.2) gives also a bound for the other direction. To be precise, we use(3.3) η i p ε q ´ ε N p ε q ď | K i T i || Λ | ď | K i || T i || Λ | and again part (b) of Lemma 3.3 to obtain | T i || Λ | ´ η i p ε q | K i | ě η i p ε q ´ ε N p ε q | K i | ´ η i p ε q | K i | “ ´ ε N p ε q | K i | ě ´ εη i p ε q | K i | .This implies the claimed bound. (cid:44) Finally, we provide a generalization of almost additivity for sets which are not disjoint, butonly ε -disjoint. The proof can be found in [13, Lemma 5.23]. Lemma 3.5.
Let G be a finitely generated group, f an admissible field with boundary term b ,and ε P p , { q . Then for any ε -disjoint sets Q i , i P t , . . . , k u , we have for each ω P Ω : (cid:13)(cid:13)(cid:13)(cid:13) f p Q, ω q ´ k ÿ i “ f p Q i , ω q (cid:13)(cid:13)(cid:13)(cid:13) ď ε p K f ` D f q | Q | ` k ÿ i “ b p Q i q ,where Q : “ Ť ki “ Q i and D f is the constant from (A3) of Definition 2.2. Approximation via the empirical measure
Given some Følner sequence p Λ j q and an admissible field f , the aim of this section is theapproximation of the expression f p Λ j ,ω q | Λ j | using elements of a second Følner sequence p Q n q andassociated empirical measures, cf. Lemma 4.3. This second sequence needs to satisfy certainadditional assumptions, namely we need that p Q n q is nested and satisfies for the correlationlength r P N from (M3) that the sequences ˆ b p Q n q | Q n | ˙ , ˆ b p Q rn q | Q n | ˙ and ˆ | B r p Q n q || Q n | ˙ converge monotonically to 0.(4.1)That these sequences converge to zero is clear by the fact that p Q n q is a Følner sequenceand b a boundary term in the sense of Definition 2.2. In order to obtain the monotonicityin (4.1), we choose a subsequence of p Q n q . These considerations show that each amenablegroup admits a nested Følner sequence p Q n q which satisfies (4.1). These terms will be used inthe error estimates in the approximations throughout this text. To abbreviate the notation,we define(4.2) β n : “ max " b p Q n q | Q n | , b p Q rn q | Q n | , | B r p Q n q || Q n | * and β p ε q : “ β ? ε ` β r {? ε s for n P N and ε P p , { q . Note that p β n q n is a monotone sequence and converges to 0, andthat by Lemma 3.3(c)(4.3) N p ε q ÿ i “ β i η i p ε q ď β p ε q ε Œ ÝÝÝÑ Remark . For the proof of Theorem 2.5, we additionally have to ensure β n ď p n q ´ for all n P N while taking the subsequences above. We will track the boundary terms throughoutthe paper and use β p ε q until the very end, where we simplify the result by applying β p ε q “ β ? ε ` β r {? ε s ď ? ε ` r {? ε s ď ? ε .The cost of this additional condition on the boundary terms is that, via Theorem 3.2, j p ε q inTheorem 2.5 will potentially increase. But up to here, we deal only with the geometry of G and still have that j p ε q depends only on ε .Moreover, let us emphasize that when considering an admissible set U the value ? ε gives auniform bound on β p ε q for all f P U , since in this situation all f P U are almost additive withthe same boundary term b . LIVENKO–CANTELLI THEORY FOR ALMOST ADDITIVE FIELDS ON AMENABLE GROUPS 11
Define for an admissible field f and Λ P F the function(4.4) f Λ : Ω Λ Ñ B , f Λ p ω q : “ f p Λ , ω q where ω P Π ´ pt ω uq .Note that by (A2) of Definition 2.2 we see that f Λ is well-defined (and measurable). In thesituation where we insert elements of the Følner sequence p Λ n q or p Λ rn q , for some r P N , wewrite(4.5) f n : “ f Λ n or f rn : “ f Λ rn .For given K, T P F and ω P Ω we define the empirical measure by(4.6) L ω p K, T q : B p Ω KT q Ñ r , s , L ω p K, T q “ | T | ÿ t P T δ p τ t ω q K .Given ε P p , { q and sequences p Λ j q and p Q j q as above, we obtain by Theorem 3.2 finitesets K i p ε q , i “ , . . . , N p ε q and (for j large enough) center sets T ji p ε q which form an ε -quasitiling of Λ j . In this setting, we use for given ω P Ω, ε P p , { q , r ą i P t , . . . , N p ε qu and j P N large enough the notation(4.7) L ωi,j p ε q : “ L ω p K i p ε q , T ji p ε qq and f i p ε q : “ f K i p ε q as well as(4.8) L r,ωi,j p ε q : “ L ω p K ri p ε q , T ji p ε qq and f ri p ε q : “ f K ri p ε q . Here, the reader may recall that K ri p ε q “ K i p ε qzB r p K i p ε qq .Moreover, we use for Λ P F , a measurable f : Ω Λ Ñ B and a measure ν on p Ω Λ , B p Ω Λ qq thenotation x f, ν y : “ ż Ω Λ f p ω q d ν p ω q . Lemma 4.2.
Let f be an admissible field and let K, T P F and ω P Ω . Then, x f K , L ω p K, T qy “ | T | ÿ t P T f p Kt, ω q .Proof. We calculate using linearity and (A1) of Definition 2.2 x f K , L ω p K, T qy “ ż Ω K f K p ω q d L ω p K, T qp ω q “ | T | ÿ t P T ż Ω K f K p ω q d δ p τ t ω q K p ω q“ | T | ÿ t P T f K pp τ t ω q K q “ | T | ÿ t P T f p Kt, ω q . (cid:44) We proceed with the first approximation Lemma.
Lemma 4.3.
Let G be a finitely generated amenable group, let f be an admissible field andlet p Λ n q and p Q n q be Følner sequences, where p Q n q is additionally nested and satisfies (4.1) .Then, we have for all ω P Ω that (4.9) lim ε Œ lim j Ñ8 (cid:13)(cid:13)(cid:13)(cid:13) f p Λ j , ω q | Λ j | ´ N p ε q ÿ i “ η i p ε q x f ri p ε q , L r,ωi,j p ε qy | K i p ε q | (cid:13)(cid:13)(cid:13)(cid:13) “ . where K i p ε q , i P t , . . . , N p ε qu are given by Theorem 3.2. Moreover, we have for arbitrary ε P p , { q and j ě j p ε q , with j p ε q from Theorem 3.2, the inequality (cid:13)(cid:13)(cid:13)(cid:13) f p Λ j , ω q | Λ j | ´ N p ε q ÿ i “ η i p ε q x f ri p ε q , L r,ωi,j p ε qy | K i p ε q | (cid:13)(cid:13)(cid:13)(cid:13) ď p K f ` D f q ε ` p ` K f ` D f q β p ε q .Proof. Let ε P p , { q and j ě j p ε q P N be given, where j p ε q is the constant given byTheorem 3.2. We estimate using the triangle inequality: (cid:13)(cid:13)(cid:13)(cid:13) f p Λ j , ω q | Λ j | ´ N p ε q ÿ i “ η i p ε q x f ri p ε q , L r,ωi,j p ε qy | K i p ε q | (cid:13)(cid:13)(cid:13)(cid:13) ď a p ε, j q ` N p ε q ÿ i “ b i p ε, j q ` N p ε q ÿ i “ c i p ε, j q , (4.10)where a p ε, j q : “ | Λ j | (cid:13)(cid:13)(cid:13)(cid:13) f p Λ j , ω q ´ N p ε q ÿ i “ ÿ t P T ji p ε q f p K i p ε q t, ω q (cid:13)(cid:13)(cid:13)(cid:13) , b i p ε, j q : “ (cid:13)(cid:13)(cid:13)(cid:13) ÿ t P T ji p ε q f p K i p ε q t, ω q | Λ j | ´ η i p ε q x f i p ε q , L ωi,j p ε qy | K i p ε q | (cid:13)(cid:13)(cid:13)(cid:13) , and c i p ε, j q : “ η i p ε q | K i p ε q | (cid:13)(cid:13) x f i p ε q , L ωi,j p ε qy ´ x f ri p ε q , L r,ωi,j p ε qy (cid:13)(cid:13) .Here, the expressions L ωi,j p ε q and f i p ε q are given by (4.7). Let us estimate the term a p ε, j q .To this end, denote the part which is covered by translates of K i p ε q , i P t , . . . , N p ε qu by R ji p ε q : “ N p ε q ď i “ K i p ε q T ji p ε q Ď Λ j .Then we have, using the properties of the ε -quasi tiling and part (a) of Lemma 3.3, | R ji p ε q | “ N p ε q ÿ i “ | K i p ε q T ji p ε q | ě | Λ j | N p ε q ÿ i “ ˆ η i p ε q ´ ε N p ε q ˙ ě p ´ ε q | Λ j | ,which in turn gives | Λ j z R ji p ε q | ď ε | Λ j | . We use this and Lemma 3.5 to calculate | Λ j | a p ε, j q ď p K f ` D f q ε | Λ j | ` b p Λ j z R ji p ε qq ` k f p Λ j z R ji p ε qq k ` N p ε q ÿ i “ ÿ t P T ji p ε q b p K i p ε q t qď p K f ` D f q ε | Λ j | ` p K f ` D f q | Λ j z R ji p ε q | ` N p ε q ÿ i “ | T ji p ε q | b p K i p ε qqď p K f ` D f q ε | Λ j | ` N p ε q ÿ i “ | T ji p ε q | b p K i p ε qq . By ε -disjointness and (3.2) we obtain12 | K i p ε q || T ji p ε q | ď p ´ ε q | K i p ε q || T ji p ε q | ď | K i p ε q T ji p ε q | ď ˆ η i p ε q ` ε N p ε q ˙ | Λ j | ,(4.11) LIVENKO–CANTELLI THEORY FOR ALMOST ADDITIVE FIELDS ON AMENABLE GROUPS 13 which together with (b) of Lemma 3.3 gives N p ε q ÿ i “ | T ji p ε q | b p K i p ε qq ď | Λ j | N p ε q ÿ i “ ˆ η i p ε q ` ε N p ε q ˙ b p K i p ε qq | K i p ε q | ď | Λ j | N p ε q ÿ i “ η i p ε q b p K i p ε qq | K i p ε q | .This implies the following bound a p ε, j q ď p K f ` D f q ε ` N p ε q ÿ i “ η i p ε q b p K i p ε qq | K i p ε q | .(4.12)To estimate the second term in (4.10), we apply Lemma 4.2 to obtain ÿ t P T ji p ε q f p K i p ε q t, ω q “ | T ji p ε q | ¨ x f i p ε q , L ωi,j p ε qy .Thus, by Corollary 3.4 and the fact k x f i p ε q , L ωi,j p ε qy k ď K f | K i p ε q | , we have for each i Pt , . . . , N p ε qu : b i p ε, j q “ (cid:13)(cid:13)(cid:13)(cid:13) | T ji p ε q | x f i p ε q , L ωi,j p ε qy | Λ j | ´ η i p ε q x f i p ε q , L ωi,j p ε qy | K i p ε q | (cid:13)(cid:13)(cid:13)(cid:13) “ (cid:12)(cid:12)(cid:12)(cid:12) | T ji p ε q || Λ j | ´ η i p ε q | K i p ε q | (cid:12)(cid:12)(cid:12)(cid:12) k x f i p ε q , L ωi,j p ε qy k ď εη i p ε q | K i p ε q | K f | K i p ε q | “ K f εη i p ε q .(4.13)Let us finally estimate the term c i p ε, j q . By Lemma 4.2 we have for each i P t , . . . , N p ε qu (cid:13)(cid:13) x f i p ε q , L ωi,j p ε qy ´ x f ri p ε q , L r,ωi,j p ε qy (cid:13)(cid:13) ď | T ji p ε q | ÿ t P T ji p ε q (cid:13)(cid:13) f p K i p ε q t, ω q ´ f p K ri p ε q t, ω q (cid:13)(cid:13) ď | T ji p ε q | ÿ t P T ji p ε q b p K ri p ε qq ` b pB r p K i p ε qq X K i p ε qq ` (cid:13)(cid:13) f pB r p K i p ε qq t X K i p ε q t, ω q (cid:13)(cid:13) ď b p K ri p ε qq ` p K f ` D f q | B r p K i p ε qq | .(4.14) Together with (4.10), the estimates for a p ε, j q in (4.12), for b i p ε, j q in (4.13) and for c i p ε, j q in (4.14) yield (cid:13)(cid:13)(cid:13)(cid:13) f p Λ j , ω q | Λ j | ´ N p ε q ÿ i “ η i p ε q x f ri p ε q , L r,ωi,j p ε qy | K i p ε q | (cid:13)(cid:13)(cid:13)(cid:13) ď p K f ` D f q ε ` N p ε q ÿ i “ η i p ε q b p K i p ε qq | K i p ε q | ` N p ε q ÿ i “ η i p ε q ˆ K f ε ` b p K ri p ε qq ` p K f ` D f q | B r p K i p ε qq || K i p ε q | ˙ ď p K f ` D f q ε ` N p ε q ÿ i “ η i p ε q b p K i p ε qq ` b p K ri p ε qq ` p K f ` D f q | B r p K i p ε qq || K i p ε q | .To verify (4.9), recall that we assumed that p Q n q satisfies (4.1). By the choice of K i p ε q inTheorem 3.2, this gives (cid:13)(cid:13)(cid:13)(cid:13) f p Λ j , ω q | Λ j | ´ N p ε q ÿ i “ η i p ε q x f ri p ε q , L r,ωi,j p ε qy | K i p ε q | (cid:13)(cid:13)(cid:13)(cid:13) ď p K f ` D f q ε ` N p ε q ÿ i “ η i p ε q b p Q i q ` b p Q ri q ` p K f ` D f q | B r p Q i q || Q i | looooooooooooooooooooooomooooooooooooooooooooooon ďp ` K f ` D f q β i ď p K f ` D f q ε ` p ` K f ` D f q β p ε q .The last inequality follows from (4.3). As this bound holds for arbitrary ε P p , { q and j ě j p ε q , this particularly proves (4.9). (cid:44) Approximation via Glivenko–Cantelli
In this section we aim to apply a multivariate Glivenko–Cantelli theorem in order to approxi-mate the empirical measure using the theoretical measure. Recall that a Glivenko–Cantellitheorem compares the empirical measure of a normalized sum of independent and identicallydistributed random variables with their distribution. At the end of this section, we will applythe following Glivenko–Cantelli theorem which was proved in [12] based on results by DeHardtand Wright, see [3, 17]. Monotone functions on R k were defined in (A4). Theorem 5.1.
Let p Ω , A , P q be a probability space and X t : Ω Ñ R k , t P N , independentand identically distributed random variables such that the distribution µ : “ P p X P ¨ q is absolutely continuous with respect to a product measure  k‘ “ µ ‘ on R k , where µ ‘ , ‘ Pt , . . . , k u , are σ -finite measures on R . For each n P N , we denote by L p ω q n : “ n ř nt “ δ X t the empirical distribution of p X t q t Pt ,...,n u . Further, fix M P R and let M : “ t g : R k Ñ R | g is monotone, and sup x P R k | g p x q | ď M u .Then, for all κ ą , there are a “ a p κ, M q ą and b “ b p κ, M q ą such that for all n P N ,there exists an event Ω κ,n,M P A with large probability P p Ω κ,n,M q ě ´ b exp p´ an q , such that LIVENKO–CANTELLI THEORY FOR ALMOST ADDITIVE FIELDS ON AMENABLE GROUPS 15 id KK r Λ Kt Kt Kt Kt Kt U t U t U t U t U t Figure 1. ε -covering and independence structure: The set Λ “ Λ j Ď G is ε -quasi covered by copies of K “ K i with centers in T “ T ji p ε q “ t t , . . . , t u .The sets U t “ U i,j,t , t P T , here marked by diagonal stripes, have at leastdistance r and satisfy | U t | ě p ´ ε q | K | . for all ω P Ω κ,n,M , we have sup g P M | x g, L p ω q n ´ µ y | ď κ .In particular, there exists a set Ω P A with P p Ω q “ and sup g P M | x g, L p ω q n ´ µ y | n Ñ8 ÝÝÝÑ forall ω P Ω . In the present situation we encounter several challanges when applying Theorem 5.1, causedby our tiling scheme. ‚ Each Λ j is tiled using N p ε q different shapes. Thus, the corresponding random variables(for different shapes) are not identically distributed . ‚ In an ε -quasi tiling, translates of the same shape K i are allowed to overlap. Thus, thecorresponding random variables are not necessarily independent .The first point can be handled by applying Glivenko–Cantelli theory for each shape K i separately. The second point is more challenging. The core of the following approach isthe “generation of independence” by resampling of the overlapping areas using conditionalprobabilities and controlling errors introduced on the altered areas with their volume. Let usexplain this in detail.Fix ε ą i P t , . . . , N p ε qu and j P N , j ě j p ε q , cf. Theorem 3.2, and consider Figure 1,which sketches a tile K “ K i , a finite set Λ “ Λ j , and the translates Kt , t P T : “ T ji p ε q , of K “ K i from an ε -quasi tiling. The sets(5.1) U i,j,t : “ p K ri t qz ` K i p T ji p ε qzt t uq ˘ Ď G , t P T ,are marked with stripes. Their distance is at least(5.2) d p U i,j,t , U i,j,t q ě d p K ri t, G z K i t q ą r , t ‰ t ,so the colors there are P -independent from each other. Unfortunately, if we take only thevalues on U i,j,t , t P T , we will end up with an independent, but not identically distributedsample. We therefore resample independent colors in K r z U i,j,t . Fortunately, the sets U i,j,t arelarge enough to compensate this small random perturbation. The following lemma specifiesthe resampling procedure. Lemma 5.2.
Let ε ą and I : “ Ť N p ε q i “ Ť j “ j p ε q tp i, j qu ˆ T ji p ε q . There exists a probabilityspace p Ω , B p Ω q , P q and random variables X, X i,j,t : Ω Ñ Ω , p i, j, t q P I , such that for all p i, j, t q P I ,(i) X and X i,j,t have distribution P ,(ii) X and X i,j,t agree on U i,j,t P -almost surely, and(iii) the random variables in the set t X i,j,t u t P T ji p ε q are P -independent.Proof. Theorem A.1 solves the problem of resampling in an abstract setting. We apply theresult here as follows. Since we use the canonical probability space in our construction, weapply Theorem A.1 with p S, S q : “ p Ω , A q , X : “ id Ω , I : “ Ť N p ε q i “ Ť j “ j p ε q tp i, j qu ˆ T ji p ε q , and Y j : “ σ p Π U j q , j P I . Theorem A.1 provides the following quantities, which we here wantto use as p Ω , A , P q : “ p Ω , A , P q , X : “ X , and X i,j,t : “ X j for all j “ p i, j, t q P I . Theproperties (i) and (ii) follow directly from Theorem A.1(i),(ii). With (5.2), Theorem A.1(iv)implies (iii). (cid:44) Next, we control the error we introduce by using our independent samples instead of thedependent ones.
Lemma 5.3.
Fix ε ą , an admissible f and U Ď K P F . For ω, ˜ ω P Ω with ω U “ ˜ ω U , wehave k f p ω, K q ´ f p ˜ ω, K q k ď b p K q ` p D f ` K f q | K z U | .In particular, in the notation from (4.4) – (4.8) and with the corresponding empirical measure L r,ωi,j p ε q : “ | T ji p ε q | ÿ t P T ji p ε q δ p τ t X i,j,t p ω qq Ki p ε q p ω P Ω q ,we have for P -almost all ω P Ω that k x f ri p ε q , L r,X p ω q i,j p ε q ´ L r,ωi,j p ε qy k ď b p K ri p ε qq ` p D f ` K f q ε | K ri p ε q | .Proof. The values of ω on U determine f p ω, K q up to k f p ω, K q ´ f p ω, U q k ď k f p ω, K q ´ f p ω, U q ´ f p ω, K z U q k ` k f p ω, K z U q k ď b p U q ` b p K z U q ` k f p ω, K z U q k ď b p U q ` p D f ` K f q | K z U | .With the fourth point in (A3), we can continue this estimate with b p U q ď b p K zp K z U qq ď b p K q ` b p K z U q ď b p K q ` D f | K z U | . LIVENKO–CANTELLI THEORY FOR ALMOST ADDITIVE FIELDS ON AMENABLE GROUPS 17
We now employ the triangle inequality to show the first claim: For ω, ˜ ω P Ω with ω U “ ˜ ω U ,we have k f p ω, K q ´ f p ˜ ω, K q k ď k f p ω, K q ´ f p ω, U q k ` k f p ˜ ω, U q ´ f p ˜ ω, K q k ď ` b p K q ` p D f ` K f q | K z U | ˘ .This calculation allows us to change ω on K z U to the independent values provided byLemma 5.2. To implement this, observe that for P -almost all ω P Ω and all i P t , . . . , N p ε qu , j P N , j ě j p ε q and t P T ji p ε q , the set U i,j,t from (5.1) exhausts K ri p ε q t up to a fractionof ε : | K ri p ε q t z U i,j,t | ď ε | K ri p ε q | . By construction, on U i,j,t , the colors are preserved: U i,j,t Ďt g P K ri p ε q t | X g p ω q “ X i,j,tg p ω qu . Together with Lemma 4.2 and the triangle inequality, thisimmediately implies for P -almost all ω P Ω that k x f ri p ε q , L r,X p ω q i,j p ε q ´ L r, ˜ ωi,j p ε qy k ď | T ji p ε q | ÿ t P T ji p ε q k f p K ri p ε q t, ω q ´ f p K ri p ε q t, X i,j,t p ω qq k ď b p K ri p ε qq ` p D f ` K f q ε | K ri p ε q | . (cid:44) The empirical measure L r,X p ω q i,j formed by independent samples should converge to P ri p ε q : “ P K ri p ε q .The following result makes this notion precise. It is the main result of this section. Proposition 5.4.
Let G be a finitely generated amenable group, let A P B p R q and p Ω : “ A G , B p Ω q , P q a probability space such that P satisfies (M1) to (M3). Moreover, let p Λ n q and p Q n q be Følner sequences, where p Q n q is nested and satisfies (4.1) . For given ε P p , { q ,let K i p ε q , i P t , . . . , N p ε qu , and j p ε q be given by Theorem 3.2. Furthermore, let U be anadmissible set of admissible fields.Then, for all κ ą , there exist a p ε, κ, K U q , b p ε, κ, K U q ą such that for all j ě j p ε q , there isan event Ω j,ε,κ,K U P B p Ω q with large probability P p Ω j,ε,κ,K U q ě ´ b p ε, κ, K U q exp p´ a p ε, κ, K U q | Λ j | q and the property that for all ω P Ω j,ε,κ,K U and f P U , it holds true that (cid:13)(cid:13)(cid:13)(cid:13) N p ε q ÿ i “ η i p ε q x f ri p ε q , L r,ωi,j p ε qy | K i p ε q | ´ N p ε q ÿ i “ η i p ε q x f ri p ε q , P ri p ε qy | K i p ε q | (cid:13)(cid:13)(cid:13)(cid:13) ď β p ε q ` p D f ` K f q ε ` κ .In particular, there is an event ˜Ω P B p Ω q with P p ˜Ω q “ such that for all ω P ˜Ω , we have lim ε Œ sup f P U (cid:13)(cid:13)(cid:13)(cid:13) N p ε q ÿ i “ η i p ε q x f ri p ε q , L r,ωi,j p ε qy | K i p ε q | ´ N p ε q ÿ i “ η i p ε q x f ri p ε q , P ri p ε qy | K i p ε q | (cid:13)(cid:13)(cid:13)(cid:13) “ . Proof.
Fix f P U . For ε P p , { q , j P N and ω P Ω, two applications of the triangle inequalitygive ∆ f p ε, ω q : “ (cid:13)(cid:13)(cid:13)(cid:13) N p ε q ÿ i “ η i p ε q x f ri p ε q , L r,ωi,j p ε qy | K i p ε q | ´ N p ε q ÿ i “ η i p ε q x f ri p ε q , P ri p ε qy | K i p ε q | (cid:13)(cid:13)(cid:13)(cid:13) ď N p ε q ÿ i “ η i p ε q | K i p ε q | (cid:13)(cid:13) x f ri p ε q , L r,ωi,j p ε q ´ P ri p ε qy (cid:13)(cid:13) ď inf ω P X ´ pt ω uq ˆ N p ε q ÿ i “ η i p ε q γ p i, j, ε, ω q ` N p ε q ÿ i “ η i p ε q γ p i, j, ε, ω q ˙ ,(5.3)where ω P Ω extends ω , i. e. X p ω q “ ω in the notation of Lemma 5.2, and γ p i, j, ε, ω q : “ (cid:13)(cid:13) x f ri p ε q , L r,ωi,j p ε q ´ L r,ωi,j p ε qy (cid:13)(cid:13) | K i p ε q | and γ p i, j, ε, ω q : “ (cid:13)(cid:13) x f ri p ε q , L r,ωi,j p ε q ´ P ri p ε qy (cid:13)(cid:13) | K i p ε q | .By Lemma 5.3 and assumption (4.1), we see that for all ω P Ω with X p ω q “ ωγ p i, j, ε, ω q ď b p K ri p ε qq | K ri p ε q | ` p D f ` K f q ε ď b p Q i q | Q i | ` p D f ` K f q ε .With Lemma 3.3(a) and (4.3), we yield the deterministic upper bound N p ε q ÿ i “ η i p ε q γ p i, j, ε, ω q ď β p ε q ` p D f ` K f q ε for all ω P X ´ p ω q Ď Ω. By now, our overall inequality (5.3) reads(5.4) ∆ f p ε, ω q ď β p ε q ` p D f ` K f q ε ` inf ω P X ´ pt ω uq N p ε q ÿ i “ η i p ε q γ p i, j, ε, ω q .To deal with γ , recall that the norm on the Banach space B our admissible fields map into isthe sup-norm. We translate the sup-norm into the Glivenko–Cantelli setting as follows. Let M f : “ t g ri,E : R | K ri p ε q | Ñ R , g ri,E p ω q : “ f ri p ω qp E q{ | K i p ε q | | E P R u .Therefore, we can write γ p i, j, ε, ω q “ sup g P M f | x g, L r,ωi,j p ε q ´ P ri p ε qy | ď sup f P U sup g P M f | x g, L r,ωi,j p ε q ´ P ri p ε qy | .From (2.5) we see that the fields in M U : “ Ť f P U M f are bounded by K U . As assumed in(A4), the fields in M U are also monotone. By Lemma 5.2(iii), the samples are independent,too. This is crucial in order to invoke Theorem 5.1. We thus obtain that, for each κ ą ε P p , { q , i P t , . . . , N p ε qu and j P N , j ě j p ε q , there are a i ” a p i, ε, κ, K U q ą b i ” b p i, ε, κ, K U q ą i,j ” Ω i,j,ε,κ,K U P B p Ω q such that P p Ω i,j q ě ´ b i exp p´ a i | T ji p ε q | q and sup ω P Ω i,j γ p i, j, ε, ω q ď κ . LIVENKO–CANTELLI THEORY FOR ALMOST ADDITIVE FIELDS ON AMENABLE GROUPS 19
We need this estimate for all i P t , . . . , N p ε qu simultaneously and considerΩ j ” Ω j,ε,κ,K U : “ N p ε q č i “ Ω i,j .To estimate the probability of Ω j is the next step. From (3.3) and Lemma 3.3(b), we notethat | T ji p ε q | ě ´ η i p ε q ´ ε N p ε q ¯ | Λ j || K i p ε q | ě p ´ ε q εN p ε q | K i p ε q | | Λ j | .With the definition a ” a ε,κ,K U : “ p ´ ε q εN p ε q min i Pt ,...,N p ε qu a i | K i p ε q | and b ” b ε,κ,K U : “ N p ε qu ÿ i “ b i ,we get P p Ω i,j q ě ´ b i exp p´ a | Λ j | q and P p Ω j q “ ´ P ´ N p ε q ď i “ Ω z Ω i,j ¯ ě ´ N p ε q ÿ i “ P p Ω z Ω i,j q ě ´ b exp p´ a | Λ j | q p Ω , B p Ω q , P q to p Ω , B p Ω q , P q . The set X p Ω j q Ď Ω seems tobe a good candidate, because for all ω P X p Ω j q , there exists ω P X ´ pt ω uq X Ş N p ε q i “ Ω i,j , andthus we can estimate inf ω P X ´ pt ω uq N p ε q ÿ i “ η i p ε q γ p i, j, ε, ω q ď N p ε q ÿ i “ η i p ε q κ ď κ .Together with (5.4), this inequality shows the claimed bound on ∆ f p ε, ω q for all ω P X p Ω j q .Unfortunately, the image of a measurable set under a measurable map is not necessarilymeasurable, but only analytic, see [2, Theorem 10.23]. At least the outer measure of ourcandidate is bounded from below by P ˚ p X p Ω j qq : “ inf B P B p Ω q ,X p Ω j qĎ B P p B q “ inf B P B p Ω q ,X p Ω j qĎ B P p X P B qě inf B P B p Ω q ,X p Ω j qĎ B P p Ω j q “ P p Ω j q ě ´ b exp p´ a | Λ j | q{ P ˚ is a nice capacity, and the Choquet CapacityTheorem [2, Theorem 10.39] states for the analytic set X p Ω j q that P ˚ p X p Ω j qq “ sup K Ď X p Ω j q compact P p K q .Thus, there exists a compact subset Ω j,ε,κ,K U Ď X p Ω j q with probability at least 1 ´ b exp p´ a | Λ j | q .We finish the proof with a standard Borel–Cantelli argument to show that ˜Ω exists as claimed.For all κ ą
0, the events A κ : “ ď n “ j p ε q 8 č j “ n Ω j,ε,κ,K U have probability 1, since ÿ j “ j p ε q P p Ω z Ω j,ε,κ,K U q ď ÿ j “ j p ε q b exp p´ a | Λ j | q ď b ÿ j “ j p ε q exp p´ a q j ă 8 .Note that by (5.4), β p ε q Ñ
0, and by construction of A k , for all ω P A κ , we havelim ε Œ sup f P U ∆ f p ε, ω q ď κ .Thus, the event ˜Ω : “ Ş k P N A { k has full probability P p ˜Ω q “
1, and for all ω P ˜Ω, we havelim ε Œ sup f P U ∆ f p ε, ω q “ (cid:44) Almost additivity and Cauchy sequences
The following calculations are devoted to a Cauchy sequence argument to obtain the desiredlimit function f ˚ . Lemma 6.1.
Let G be a finitely generated amenable group, let A P B p R q and p Ω “ A G , B p Ω q , P q a probability space such that P satisfies (M1) to (M3). Moreover, let f bean admissible field and p Q n q a nested Følner sequence satisfying (4.1) . Then, there exists f ˚ P B with lim ε Œ (cid:13)(cid:13)(cid:13)(cid:13) N p ε q ÿ i “ η i p ε q x f ri p ε q , P ri p ε qy | K i p ε q | ´ f ˚ (cid:13)(cid:13)(cid:13)(cid:13) “ ,where for k P N and ε P p {p k ` q , { k q the sets K i p ε q , i P t , . . . , N p ε qu are extracted fromthe sequence p Q n ` k q n via Theorem 3.2. The approximation error is bounded by (cid:13)(cid:13)(cid:13)(cid:13) N p ε q ÿ j “ η j p ε q x f rj p ε q , P rj p ε qy | K j p ε q | ´ f ˚ (cid:13)(cid:13)(cid:13)(cid:13) ď p K f ` D f q ε ` p ` K f ` D f q β p ε q .Proof. In order to prove the existence of f ˚ , we study for ε, δ P p , { q the difference D p ε, δ q : “ (cid:13)(cid:13)(cid:13)(cid:13) N p ε q ÿ j “ η j p ε q x f rj p ε q , P rj p ε qy | K j p ε q | ´ N p δ q ÿ i “ η i p δ q x f ri p δ q , P ri p δ qy | K i p δ q | (cid:13)(cid:13)(cid:13)(cid:13) . Our aim is to show lim δ Œ lim ε Œ D p ε, δ q “
0. To prove this, we insert terms which interpolatebetween the minuend and the subtrahend. These terms will be given using Theorem 3.2. Foreach ε P p {p k ` q , { k s , we apply Theorem 3.2 to choose the sets K j p ε q , j “ , . . . , N p ε q ,from the Følner sequence p Q n ` k q n P N . The particular choice of the sets K j p ε q , j “ , . . . , N p ε q ,as elements of the sequence p Q n ` k q n ensures that for given δ ą ε ą ε P p , ε q each K j p ε q , j “ , . . . , N p ε q , can be δ -quasi tiled with the elements K i p δ q , i “ , . . . , N p δ q . As in Theorem 3.2, we denote the associated center sets by T ji p δ q , where weemphasize the dependence on the parameter δ .For K P F we use the notation F p K q : “ x f K , P K y (6.1)and hence for the tiles K j p ε q , i “ , . . . , N p ε q , we write F p K ri p ε qq : “ x f ri p ε q , P ri p ε qy . Thefunction F is translation invariant, i. e. for all K P F and t P G we have F p Kt q “ F p K q . LIVENKO–CANTELLI THEORY FOR ALMOST ADDITIVE FIELDS ON AMENABLE GROUPS 21
With the convention (6.1) and using the triangle inequality we obtain D p ε, δ q ď D p ε, δ q ` D p ε, δ q , where D p ε, δ q : “ (cid:13)(cid:13)(cid:13)(cid:13) N p ε q ÿ j “ η j p ε q F p K rj p ε qq ´ ř N p δ q i “ | T ji p δ q | F p K ri p δ qq | K j p ε q | (cid:13)(cid:13)(cid:13)(cid:13) , and D p ε, δ q : “ (cid:13)(cid:13)(cid:13)(cid:13) N p ε q ÿ j “ η j p ε q ř N p δ q i “ | T ji p δ q | F p K ri p δ qq | K j p ε q | ´ N p δ q ÿ i “ η i p δ q F p K ri p δ qq | K i p δ q | (cid:13)(cid:13)(cid:13)(cid:13) . The translation invariance of F and the triangle inequality yield(6.2) D p ε, δ q ď N p ε q ÿ j “ η j p ε q | K j p ε q | (cid:13)(cid:13)(cid:13)(cid:13) F p K rj p ε qq ´ N p δ q ÿ i “ ÿ t P T ji p δ q F p K ri p δ q t q (cid:13)(cid:13)(cid:13)(cid:13) . We decompose K rj p ε q in the following way K rj p ε q “ N p δ q ď i “ ď t P T ji p δ q K ri p δ q t Y K rj p ε qz N p δ q ď i “ K i p δ q T ji p δ q Y Y ˜ˆ K rj p ε qz N p δ q ď i “ K ri p δ q T ji p δ q ˙ X N p δ q ď i “ ` K i p δ q X B r p K i p δ qq ˘ T ji p δ q ¸ “ : α Y α Y α .By definition of the function F the almost additivity of the admissible field f inherits to F .Note that δ -disjointness of the sets K i t , t P T ji p δ q implies δ -disjointness of the sets K ri t , t P T ji p δ q . Therefore, applying almost additivity, Lemma 3.5 and the properties of admissiblefields and the boundary term we obtain (cid:13)(cid:13)(cid:13)(cid:13) F p K rj p ε qq ´ N p δ q ÿ i “ ÿ t P T ji p δ q F p K ri p δ q t q (cid:13)(cid:13)(cid:13)(cid:13) ď (cid:13)(cid:13)(cid:13)(cid:13) F p K rj p ε qq ´ ÿ i “ F p α i q (cid:13)(cid:13)(cid:13)(cid:13) ` (cid:13)(cid:13)(cid:13)(cid:13) F p α q ´ N p δ q ÿ i “ ÿ t P T ji p δ q F p K ri p δ qq (cid:13)(cid:13)(cid:13)(cid:13) ` k F p α q k ` k F p α q k ď ÿ i “ b p α i q ` δ p K f ` D f q | K j p ε q | ` N p δ q ÿ i “ ÿ t P T ji b p K ri p δ qq ` K f | α | ` K f | α | ď δ p K f ` D f q | K j p ε q | ` N p δ q ÿ i “ ÿ t P T ji p δ q b p K ri p δ qq ` p K f ` D f q | α | ` p K f ` D f q | α | .Next, we estimate the sizes of α and α . For α we drop some of the intersections in itsdefinition. In order to give a bound on the size of α , we use that K rj p ε q is p ´ ε q -coveredby t K ri p δ q | i u , more specifically, part (iii) in Definition 3.1. We obtain | α | ď δ | K j p ε q | and | α | ď N p δ q ÿ i “ | T ji p δ q || B r p K i p δ qq | , and therewith achieve (cid:13)(cid:13)(cid:13)(cid:13) F p K rj p ε qq ´ N p δ q ÿ i “ ÿ t P T ji p δ q F p K ri p δ q t q (cid:13)(cid:13)(cid:13)(cid:13) ď δ p K f ` D f q | K j p ε q | ` N p δ q ÿ i “ | T ji p δ q | ´ b p K ri p δ qq ` p K f ` D f q | B r p K i p δ qq | ¯ . This together with (6.2) and part (a) of Lemma 3.3 yields D p ε, δ qď N p ε q ÿ j “ ˆ δ p K f ` D f q η j p ε q ` N p δ q ÿ i “ η j p ε q | T ji p δ q || K j p ε q | ´ b p K ri p δ qq ` p K f ` D f q | B r p K i p δ qq | ¯˙ ď δ p K f ` D f q ` N p ε q ÿ j “ N p δ q ÿ i “ η j p ε q | T ji p δ q || K j p ε q | ´ b p K ri p δ qq ` p K f ` D f q | B r p K i p δ qq | ¯ . As δ is assumed to be smaller than 1 {
10, we can apply Corollary 3.4, which gives for arbitrary i P t , . . . , N p δ qu and j P t , . . . , N p ε qu | T ji p δ q || K j p ε q | ď η i p δ q | K i p δ q | ` δη i p δ q | K i p δ q | ď η i p δ q | K i p δ q | .Inserting this in the last estimate for D p ε, δ q implies together with part (a) of Lemma 3.3that D p ε, δ q ď δ p K f ` D f q ` N p δ q ÿ i “ η i p δ q | K i p δ q | ´ b p K ri p δ qq ` p K f ` D f q | B r p K i p δ qq | ¯ . Now, we use the monotonicity assumption in (4.1), which allows to replace the elements K ri p δ q and K i p δ q by Q ri and Q i , respectively: D p ε, δ q ď δ p K f ` D f q ` N p δ q ÿ i “ η i p δ q | Q i | ´ b p Q ri q ` p K f ` D f q | B r p Q i q | ¯ . (6.3)Let us proceed with the estimation of D p ε, δ q : D p ε, δ q “ (cid:13)(cid:13)(cid:13)(cid:13) N p δ q ÿ i “ F p K ri p δ qq ˆ N p ε q ÿ j “ η j p ε q | T ji p δ q || K j p ε q | ´ η i p δ q | K i p δ q | ˙ (cid:13)(cid:13)(cid:13)(cid:13) . (6.4)With the triangle inequality, Corollary 3.4, and part (a) of Lemma 3.3 we obtain (cid:12)(cid:12)(cid:12)(cid:12) N p ε q ÿ j “ η j p ε q | T ji p δ q || K j p ε q | ´ η i p δ q | K i p δ q | (cid:12)(cid:12)(cid:12)(cid:12) ď N p ε q ÿ j “ η j p ε q (cid:12)(cid:12)(cid:12)(cid:12) | T ji p δ q || K j p ε q | ´ η i p δ q | K i p δ q | (cid:12)(cid:12)(cid:12)(cid:12) ` (cid:12)(cid:12)(cid:12)(cid:12) N p ε q ÿ j “ η j p ε q ´ (cid:12)(cid:12)(cid:12)(cid:12) η i p δ q | K i p δ q | ď N p ε q ÿ j “ η j p ε q δη i p δ q | K i p δ q | ` εη i p δ q | K i p δ q | ď δη i p δ q | K i p δ q | ` εη i p δ q | K i p δ q | . LIVENKO–CANTELLI THEORY FOR ALMOST ADDITIVE FIELDS ON AMENABLE GROUPS 23
This together with (6.4) gives the bound D p ε, δ q ď N p δ q ÿ i “ K f | K ri p δ q | ˆ δη i p δ q | K i p δ q | ` εη i p δ q | K i p δ q | ˙ ď K f δ ` K f ε. (6.5)Thus, the estimates of D p ε, δ q and D p ε, δ q in (6.3) and (6.5) together yield D p ε, δ q ď K f ε ` δ p K f ` D f q ` N p δ q ÿ i “ η i p δ q | Q i | ` b p Q ri q ` p K f ` D f q | B r p Q i q | ˘ (6.6)for all δ ą ε P p , ε p δ qq . Applying part (c) of Lemma 3.3 we seelim δ Œ lim ε Œ D p ε, δ q “ B is a Banach space we obtain that there existsan element f ˚ P B with lim ε Œ (cid:13)(cid:13)(cid:13)(cid:13) N p ε q ÿ j “ η j p ε q x f rj p ε q , P rj p ε qy | K j p ε q | ´ f ˚ (cid:13)(cid:13)(cid:13)(cid:13) “ . In order to get the error estimate for finite δ ą
0, we use (6.6), Lemma 3.3(c), and (4.1) asfollows (cid:13)(cid:13)(cid:13)(cid:13) N p δ q ÿ j “ η j p δ q x f rj p δ q , P rj p δ qy | K j p δ q | ´ f ˚ (cid:13)(cid:13)(cid:13)(cid:13) “ lim ε Œ D p ε, δ qď p K f ` D f q δ ` N p δ q ÿ i “ η i p δ q | Q i | ` b p Q ri q ` p K f ` D f q | B r p Q i q | ˘ ď p K f ` D f q δ ` p ` K f ` D f q β p δ q . (cid:44) Proof of the main theorem
We will prove a slightly more explicit statement which tracks the geometric error in termsof ε and the probabilistic error in terms of κ separately. Theorem 2.5 is implied by the choice κ : “ ? ε . Recall that B is the Banach space of bounded and right-continuous functions from R to R . Theorem 7.1.
Let G be a finitely generated amenable group. Further, let A P B p R q and p Ω “ A G , B p Ω q , P q a probability space such that P satisfies (M1) to (M3). Finally, let U be anadmissible set of admissible fields with common bound K U , cf. Definition 2.2.Then, there exists a limit element f ˚ P B with the following properties. For each Følnersequence p Λ n q , ε P p , { q and κ ą , there exist j p ε q P N , which is independent of κ and K U , and a p ε, κ, K U q , b p ε, κ, K U q ą , such that for all j P N , j ě j p ε q , there is an event Ω j,ε,κ,K U P B p Ω q with the properties P p Ω j,ε,κ,K U q ě ´ b p ε, κ, K U q exp ` ´ a p ε, κ, K U q | Λ j | ˘ and (cid:13)(cid:13)(cid:13)(cid:13) f p Λ j , ω q | Λ j | ´ f ˚ (cid:13)(cid:13)(cid:13)(cid:13) ď p K f ` D f ` q? ε ` κ for all ω P Ω j,ε,κ,K U and all f P U . Proof.
We follow the path prescribed in the previous chapters and ‚ quasi tile Λ j , j ě j p ε q , with K i p ε q , i “ , . . . , N p ε q , see Theorem 3.2, ‚ approximate | Λ j | ´ f p Λ j , ω q with the empirical measures L r,ωi,j p ε q , cf. (4.8) and Lemma 4.3, ‚ express the empirical measures by their limiting counterparts P ri p ε q with Proposi-tion 5.4, and ‚ use the Cauchy property of the remaining terms to obtain a limiting function f ˚ , seeLemma 6.1.To confirm the error estimate, we employ the triangle inequality (cid:13)(cid:13)(cid:13)(cid:13) f p Λ j , ω q | Λ j | ´ f ˚ (cid:13)(cid:13)(cid:13)(cid:13) ď (cid:13)(cid:13)(cid:13)(cid:13) f p Λ j , ω q| Λ j | ´ N p ε q ÿ i “ η i p ε q x f ri p ε q , L r,ωi,j p ε qy | K i p ε q | (cid:13)(cid:13)(cid:13)(cid:13) ` (cid:13)(cid:13)(cid:13)(cid:13) N p ε q ÿ i “ η i p ε q x f ri p ε q , L r,ωi,j p ε qy | K i p ε q | ´ N p ε q ÿ i “ η i p ε q x f ri p ε q , P ri p ε qy | K i p ε q | (cid:13)(cid:13)(cid:13)(cid:13) ` (cid:13)(cid:13)(cid:13)(cid:13) N p ε q ÿ i “ η i p ε q x f ri p ε q , P ri p ε qy | K i p ε q | ´ f ˚ (cid:13)(cid:13)(cid:13)(cid:13) “ : ∆ p ε, j, ω q .By Lemmas 6.1 and 4.3 and Proposition 5.4, we immediately get that there is an event˜Ω P B p Ω q with full probability P p ˜Ω q “ ε Œ lim j Ñ8 ∆ p ε, j, ω q “ ω P ˜Ω.Furthermore, Proposition 5.4 provides the event Ω j,ε,κ,K U with probability as large as claimed,and by collecting all the error terms and by Remark 4.1, we see that for all ε P p , { q , j ě j p ε q , κ ą f P U , and ω P Ω j,ε,κ,K U , see Proposition 5.4, (cid:13)(cid:13)(cid:13)(cid:13) f p Λ j , ω q | Λ j | ´ f ˚ (cid:13)(cid:13)(cid:13)(cid:13) ď p K f ` D f q ε ` p K f ` D f ` q β p ε q ` κ ď p K f ` D f ` q? ε ` κ .Note the uniformity of the last inequality for all f P U is also discussed in Remark 4.1.To see that the limit f ˚ does not depend on the specific choice of p Λ j q use the followingargument: Every two Følner sequences can be combined two one Følner sequence, whichyields by our theory a limit f ˚ P B . As the two original sequences are subsequences, they leadto the same limit function f ˚ . (cid:44) Appendix A. Conditional resampling
In Lemma 5.2, we need to remove the dependent parts of samples. We achieve this by resamplingthe critical parts of the samples, keeping the large enough already independent parts. This is doneby augmenting the probability space to provide room for more random variables. The problem ofresampling turned out to be treatable in a much broader setting, so a general tool is provided here.
Theorem A.1 (Resampling) . Let p Ω , A , P q be a Borel probability space, p S, S q a Borel space, and X : Ω Ñ S an S -valued random variable with distribution P X : “ P ˝ X ´ : S Ñ r , s . Further let I bean index set, and for each j P I , let Y j Ď S be a σ -algebra.Then, there is a probability space p Ω , A , P q such that for all j P I , maps as indicated in the followingdiagram exist and are measure preserving, and all the diagrams commute almost surely. LIVENKO–CANTELLI THEORY FOR ALMOST ADDITIVE FIELDS ON AMENABLE GROUPS 25 p Ω , A , P qp S, Y j , P X | Y j q p S, S , P X qp S, S , P X qp Ω , A , P q X Π X X j id S id S This means in particular that Π is measure preserving, and that, for all j P I ,(i) the random variable X j has distribution P X ,(ii) for each measure space p T, T q and each Y j - T -measurable map g : p S, Y j q Ñ p T, T q , we have g p X q “ g p X j q P -almost surely.Furthermore, the joint distribution of p X j q j P I has the following properties.(iii) For each finite subset F Ď I and A F “ Ś j P F A j , where A j P S , we have P X -almost surelythat P p X F P A F | X “ ¨ q “ ź j P F P p X j P A j | X “ ¨ q “ ź j P F P X p A j | Y j q .In particular, the random variables X j , j P I , are independent when conditioned on X .(iv) If, for a (not necessarily finite) subset J Ď I , the σ -algebras Y j , j P J , are P X -independent,then the random variables X j , j P J , are P -independent. Since Π is measure preserving, p Ω , A , P q extends p Ω , A , P q . Property (i) justifies the name resampling.Statement (ii) says that in X j the information contained in Y j is preserved throughout the resampling, j P I . Point (iii) states that the new random variables copied only the information from Y j , j P I , andnot more. In (iv), we learn how to provide independence of the resampling random variables. Proof.
We define the spaces and maps as follows:Ω : “ Ω ˆ S I , A : “ A b S b I ,Π : Ω Ñ Ω, Π p ω, p s j q j P I q : “ ω , X : Ω Ñ S , X p ω, p s j q j P I q : “ X p ω q , X j : Ω Ñ S , X j p ω, p s k q k P I q : “ s j We now define the measure P via Kolmogorov’s extension theorem, see [4, Theorem 14.36]. We need aconsistent family of probability measures. For a more unifying notation, we augment I : “ t u Y I .Fix a finite subset F Ď I . If 0 P F , we define a probability measure P F : A b S b F zt u Ñ r , s . In case0 R F , we define a probability measure P F : S b F Ñ r , s . If 0 P F , then choose A P A , otherwise, let A : “ Ω. For all j P F zt u we let A j P S . Now let A F : “ Ś j P F A j and(A.1) P F p A F q : “ E ” A ź j P F zt u P X p A j | Y j q ˝ X ı .Here, E denotes integration with respect to P . By the extension theorem for measures, see [4,Theorem 1.53], (A.1) defines a probability measure. The family p P F q F Ď I finite is consistent. Forexample, for finite subsets 0 R F Ď J Ď I with the projection Π JF : S J Ñ S F and A F “ Ś j P F A j with A j P S , we have p Π JF q ´ p A F q “ A F ˆ Ś j P J z F S . Thus, P J ` p Π JF q ´ p A F q ˘ “ E X ”ź j P F P X p A j | Y j q ź j P J z F P X p S | Y j q ı “ P F p A F q ,where E X is integration with respect to P X . The remaining cases 0 P F Ď J , and 0 R F but 0 P J workanalogously. By Kolmogorov’s extension theorem, we have exactly one measure P : “ lim ÐÝ F Ď I P F : A Ñr , s . We now verify the properties of P . Let us first check, that Π is measure preserving. Indeed, for A P A ,we have P p Π P A q “ P t u p A q “ E r A s “ P p A q .Now we already know that X “ X ˝ Π is measure preserving, too.Ad (i): For all j P I and B P S , we have P p X j P B q “ P t j u p B q “ E X r P X p B | Y j qs “ E X r B s “ P X p B q .Ad (ii): Let j P I , p T, T q be a measure space and g : S Ñ T be Y j - T -measurable. We determine thejoint distribution of X and X j . By (A.1), we have, for B, B P T , that A : “ g ´ p B q P Y j as well as A : “ g ´ p B q P Y j , and P p g p X q P B, g p X j q P B q “ P p X P A, X j P A q “ P t ,j u p X ´ p A q ˆ A q“ E r X ´ p A q P X p A | Y j q ˝ X s “ E X r A A s“ P X p A X A q “ P p X P A X A q “ P p g p X q P B X B q ,(A.2)where in the last line, we used that A X A “ g ´ p B q X g ´ p B q “ g ´ p B X B q . Now, since therectangles t B ˆ B | B, B P T u are stable under intersections and generate T b T , equation (A.2)determines the distribution of p g p X q , g p X j qq : Ω Ñ T . Note, that the measure which is concentratedon the diagonal tp t, t q | t P T u with both marginals equal to P X ˝ g ´ satisfies (A.2), too. Therefore, P p g p X q “ g p X j qq “ F Ď I and A j P S for j P F , and let A F : “ Ś j P F A j . For all B P S , wehave E r t X P B u P p X F P A F | X qs “ E r t X P B u E r t X F P A F u | X ss “ E r t X P B u t X F P A F u s“ P r X P B, X F P A F s “ P t uY F p X ´ p B q ˆ A F q“ E ” X ´ p B q ź j P F P X p A F | Y j q ˝ X ı “ E ” t X P B u ź j P F P X p A F | Y j q ˝ X ı .Since σ p X q “ tt X P B u | B P S u , this proves P p X F P A F | X q “ ź j P F P X p X j P A j | Y j q ˝ X P -almost surely. For F “ t j u , we get P p X j P A j | X q “ P X p X j P A j | Y j q , too. The claim is thefactorized version of these statements, which exist because p S, S q is a Borel space.Ad (iv): For F Ď J finite and A F “ Ś j P F A j with A j P S , we use (iii) to get P p X F P A F q “ E r P p X F P A F | X qs“ E ”ź j P F P X p A j | Y j q ˝ X ı “ E X ”ź j P F P X p A j | Y j q ı .The σ -algebras Y j , j P F Ď J , are P X -independent. This independence is inherited by Y j -measurablefunctions like P X p A j | Y j q . We can therefore continue the calculation with P p X F P A F q “ ź j P F E X “ P X p A j | Y j q ‰ “ ź j P F P X p A j q “ ź j P F P p X j P A j q .Since the cylinder sets generate S b J , this is the claimed P -independence. (cid:44) LIVENKO–CANTELLI THEORY FOR ALMOST ADDITIVE FIELDS ON AMENABLE GROUPS 27
Appendix B. Proof summary for montilable amenable groups
The proofs of [12] concerning the case G “ Z d can be generalized to apply to a finitely generatedamenable group G if it satisfies the tiling property ( ‘ ).We list the major changes which are necessary for this purpose:(a) Instead of defining the set T m,n using multiples of m (c. f. eq. (3.1) in [12]), we employ thegrid T m , namely we set T m,n : “ t t P T m | Λ m t Ď Λ n u (B.1) Thus, T m,n contains the elements of T m which correspond to translates of Λ m which arecompletely contained in Λ n . Using this definition, the empirical measures are L ωm,n and L ω,rm,n are given accordingly.(b) One needs to verify the following basic result. Given a tiling Følner sequence p Λ n q , we have(i) for each m P N the sequence p Λ m T m,n q n P N is a Følner sequence;(ii) for each m, n P N we have Λ n Ď B ρ p m q p Λ n q Y Λ m T m,n , where ρ p m q “ diam p Λ m q ; and(iii) for each m P N we have lim n Ñ8 | Λ n | { | T m,n | “ | Λ m | .(c) Points (a) and (b) allow to prove an equivalent version of Lemma 3.2 of [12] in the situationof amenable groups with property ( ‘ ), by following exactly the steps of the proof presentedtherein.(d) Besides Lemma 3.2. also Lemma 6.1. needs to be slightly changed. In fact, again by using (a)and (b) the proof can directly be adapted to the situation where G is amenable and p Λ n q is atiling Følner sequence.(e) In the end, the proof of the main theorem reduces basically to an application of the triangleinequality, the new versions of Lemma 3.2 and Lemma 6.1 as well as (the original versionof) Theorem 5.1. Note that Theorem 5.1 need not to be adapted as it is independent of thegeometry. References [1] Adachi, T. (1993). A note on the folner condition for amenability.
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Fakultät für Mathematik, TU Dortmund, 44221 Dortmund, Germany (FS)(FS)