Hyperfinite measure-preserving actions of countable groups and their model theory
aa r X i v : . [ m a t h . D S ] O c t Hyperfinite measure-preserving actions of countable groups and theirmodel theory
Pierre GiraudOctober 18, 2019
Abstract
We give a shorter proof of a theorem of G. Elek stating that two hyperfinite measure-preservingactions of a countable group on standard probability spaces are approximately conjugate if and onlyif they have the same invariant random subgroup.We then use this theorem to study model theory of hyperfinite measure-preserving actions ofcountable groups on probability spaces. This work generalizes the model-theoretic study of auto-morphisms of probability spaces conducted by I. Ben Yaacov, A. Berenstein, C. W. Henson andA. Usvyatsov.
Contents A θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Completeness and Model Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Elimination of quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6 Stability and Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 INTRODUCTION Classical ergodic theory consists of the study of probability measure-preserving (pmp in short) trans-formations of a probability space. A pmp transformation T of a probability space ( X, µ ) is abimeasurable permutation of X such that for all measurable subsets A of X , µ ( T − A ) = µ ( A ). It iscalled ergodic if any T -invariant subset of X is either null or conull, and it is called aperiodic ifalmost every T -orbit is infinite. In the case of a single transformation T of an atomless probabilityspace, it is well-known that ergodicity implies aperiodicity. For now, we restrict ourselves to standardprobability spaces , that is probability spaces that are isomorphic to the interval [0 ,
1] equipped withthe Lebesgue measure.Two pmp transformations T and T ′ are said to be conjugate , or sometimes isomorphic , if thereis a third pmp transformation S such that up to a null set, T ′ = ST S − . One of the main goals ofergodic theory is to understand the conjugacy relation on pmp transformations, particularly on theset of ergodic pmp transformations. Conjugacy is completely understood in some specific cases, forexample, entropy is a complete invariant of conjugacy for Bernoulli shifts [Orn70] and spectrum is acomplete invariant of conjugacy for compact transformations. However, in general, conjugacy is a verycomplicated relation as shown in [FW04] and [FRW11].In this paper we study the simpler relation of approximate conjugacy. Two pmp transformations T and T ′ of ( X, µ ) are said to be approximately conjugate if for all ε > S of ( X, µ ) such that T ′ = ST S − up to a set of measure at most ε . It is a well-knownconsequence of Rokhlin Lemma that any two aperiodic pmp transformations of standard probabilityspaces are approximately conjugate [Kec10, Thm. 2.4]. We thus focus on understanding the approxi-mate conjugacy relation for general pmp actions of countable discrete groups rather than single pmptransformations, which correspond to Z -actions.A pmp action of a countable group Γ on a probability space ( X, µ ) is an action of Γ on X by pmptransformations. For a pmp action Γ α y ( X, µ ) and γ ∈ Γ, we let γ α denote the pmp transformationassociated to γ in the action α . Two pmp actions α and β of a countable group Γ are conjugate ifthere is a pmp transformation S such that S − γ α S = γ β for all γ ∈ Γ. We say that α is a factor of β , denoted by α ⊑ β if there is a measure-preserving map S : X → X such that γ α S = Sγ β for every γ ∈ Γ.We say that α and β are approximately conjugate if for every finite F ⊆ Γ and every ε > S of X such that µ (cid:16) { x ∈ X : ∃ γ ∈ F, γ β x = Sγ α S − x } (cid:17) < ε. This notion of approximate conjugacy comes from the study of the spaces Aut(
X, µ ) and A (Γ , X, µ )of pmp transformations of ( X, µ ) and of pmp actions of Γ on (
X, µ ), respectively.The space Aut(
X, µ ) can be equipped with two topologies: the weak and the uniform topology(see [Kec10] for definitions). Two pmp transformations T and S are called weakly equivalent if[ T ] w = [ S ] w , where [ T ] is the conjugacy class of T , and A w denotes the closure of A in the weaktopology. Then, the space of actions can be seen as a closed subspace of Aut( X, µ ) Γ equipped witheither product topology, and this induces two topologies on A (Γ , X, µ ), that we respectively call againthe weak and the uniform topology. In the same fashion as for transformations, we say that two actions α and β are weakly equivalent if [ α ] w = [ β ] w .Now approximate conjugacy is the uniform counterpart of weak equivalence, that is, two pmp ac-tions α and β are approximately conjugate if and only if [ α ] u = [ β ] u , where A u is the uniform closure of A . The study of approximate conjugacy in the present paper was mostly motivated by similar resultsobtained for weak equivalence by R. Tucker-Drob in [Tuc15].The first obstacle to approximate conjugacy is freeness : a pmp action of Γ is free if the set offixed points of any nontrivial element of Γ is null. For Z -actions, freeness corresponds to aperiodicity.It is easy to see that approximate conjugacy preserves the freeness of the actions, and that the trivialaction is only approximately conjugate with itself. INTRODUCTION α y ( X, µ ), the pushforward of the measure µ by the stabilizer application x ∈ X Stab α ( x ) gives a measure θ α on the space of subgroups of Γ. Wecall this measure the Invariant Random Subgroup (IRS in short, see [AGV14]) of the action α . Thenit is not hard to see that the IRS is an invariant of approximate conjugacy. Moreover, free actionscorrespond to the case where the IRS is the Dirac measure on the trivial subgroup δ { e } and the trivialaction corresponds to the case where the IRS is δ Γ .In this paper we work with hyperfinite actions, which are defined as follows: Definition.
A pmp action Γ y ( X, µ ) is said to be hyperfinite if for any finite subset S of Γ andany ε >
0, there exists a finite group G acting in a measure-preserving way on ( X, µ ) such that µ ( { x ∈ X : S · x ⊆ G · x } ) > − ε. It is a theorem of D. S. Ornstein and B. Weiss [OW80] that pmp actions of amenable groups arehyperfinite.In general, we have the following implications:approximate conjugacy = ⇒ weak equivalence = ⇒ same IRS . In the most general context, the IRS of an action is not a complete invariant of approximate conjugacy.However, G. Elek proved that when restricted to hyperfinite actions, it is:
Theorem A (G.Elek, [Ele12, Thm. 9]) . Let α and β be two pmp hyperfinite actions of a group Γ ona standard probability space such that θ α = θ β . Then α and β are approximately conjugate. This theorem thus generalizes the consequence [Kec10, Thm. 2.4] of Rokhlin Lemma, which canbe obtained by taking Γ = Z and θ α = θ β = δ { e } .In this paper, we give a shorter proof of this theorem, first by considering the critical case of ac-tions which are factors one of another and then using a confluence argument to conclude in the generalcase. Moreover, when one of the actions is a factor of the other, we add a slight improvement to thetheorem by requiring that the pmp transformations witnessing approximate conjugacy stabilize somemeasurable sets. This stronger version of the theorem will be used for the model theoretic study ofpmp actions, which is the main topic of the present paper.The formalism of continuous model theory that we use was developed by I. Ben Yaacov and A. Usvy-atsov.While classical model theory is concerned with algebraic theories such as discrete groups, alge-braically closed or real closed fields, its continuous counterpart allows the study of metric structures.In recent years, continuous model theory has been used to study theories such as metrics spaces, Ba-nach spaces, Hilbert spaces and measure algebras. More precisely, a particular attention was given tothe study of formulas involving automorphisms of the latter theories.In the present paper we are interested in the model theory of a group action on a probability space,in other words, we look at formulas involving finite subsets of automorphisms of a probability space( X, µ ) from a given subgroup of the group of automorphisms of (
X, µ ). However, probability spacesdo not admit a model theoretic treatment as such, where the elements of a structure are the points inprobability spaces.In order to solve this issue, we consider as structures not the probability spaces themselves buttheir associated measure algebra. For a probability space ( X, Σ , µ ), its associated measure algebraMAlg( X, µ ) is the quotient set Σ / N where N denotes the σ -ideal of null sets. It inherits the Booleanoperations ∨ , ∩ , . − of Σ and is endowed with a natural metric d µ ( π ( A ) , π ( B )) := µ ( A △ B ), where π is the quotient map. THE GENERALIZATION OF ROKHLIN LEMMA Z and the more general case of free actions of amenable groups treated byA. Berenstein and C. W. Henson in an unpublished paper.Without loss of generality, we restrict our study to actions of the free group over an infinite countablesubset, F ∞ , as any action of a countable group can be seen as an action of F ∞ . Then one can see thatthe equivalence relation of elementary equivalence is weaker than approximate conjugacy but strongerthan weak equivalence. This result highlights the link between model theory and the equivalencerelations usually studied in ergodic theory.For any IRS θ on F ∞ , we define a theory A θ axiomatizing pmp actions with IRS θ . By a result ofG. Elek ([Ele12, Thm. 2]), the hyperfiniteness of an action is determined by its IRS. We thus call anIRS θ hyperfinite if actions with IRS θ are hyperfinite.By Theorem A, in the context of hyperfinite actions, having the same IRS is equivalent to beingelementarily equivalent. We prove: Theorem B. If θ is a hyperfinite IRS, then the theory A θ is complete and model complete. However, unlike in [BYBHU08, Section 18] these theories do not admit quantifier elimination ingeneral. We nevertherless prove in Theorem 3.28 that there is a reasonable expansion of the theorywhich eliminates quantifier, and we then use this to prove
Theorem C. If θ is a hyperfinite IRS, then the theory A θ is stable and the stable independencerelation given by non dividing admits a natural characterization in terms of the classical probabilisticindependence of events (in a sense described in Definition 3.34).Acknowledgments: I am very grateful to my PhD advisors François Le Maître and Todor Tsankovfor suggesting the subject of this paper and for their valuable advice throughout the preparation andwriting of this article. I would also like to thank Tomás Ibarlucía and Robin Tucker-Drob for manyhelpful discussions and suggestions.
Definition 2.1. A graph G is a pair (V( G ) , E( G )) where V( G ) is a set and E( G ) is an irreflexive andsymmetric binary relation on V( G ). Elements of V( G ) are called vertices of G and elements of E( G )are called edges of G .For G a graph, for each v ∈ V( G ) we let deg G ( v ) = |{ u ∈ V( G ) : ( v, u ) ∈ E( G ) }| and we callsup v ∈ V( G ) deg G ( v ) ∈ N ∪ {∞} the degree bound of G . Definition 2.2. An isomorphism between the graphs G and H is a bijection f : V( G ) → V( H )such that ∀ x, y ∈ V( G ), ( x, y ) ∈ E( G ) ⇔ ( f ( x ) , f ( y )) ∈ E( H ). Definition 2.3.
Let G be a graph, A ⊆ V( G ) and B ⊆ E( G ). Then we define : • V G inc ( B ) = { v ∈ V( G ) : ∃ u ∈ V( G ) , ( u, v ) ∈ B ∨ ( v, u ) ∈ B } the set of vertices incident to B . • E G inc ( A ) = { ( a, v ) ∈ E( G ) : a ∈ A } the set of edges incident to A .We will write V inc ( B ) and E inc ( A ) when the context makes clear which graph G is considered. THE GENERALIZATION OF ROKHLIN LEMMA Definition 2.4.
Let G be a graph. A subgraph of G is a graph H such that V( H ) = V( G ) andE( H ) ⊆ E( G ). In this case, we write H ⊆ G .If V ⊆ V( G ), the subgraph of G induced by V is the graph (V( G ) , E( G ) ∩ V × V ). Nevertheless,in many cases it will be convenient to identify the induced graph on V and the graph ( V, E( G ) ∩ V × V )and therefore see the induced graph on V as a graph on the set of vertices V .In general, we write G ≃ H to indicate that G and H are isomorphic. Definition 2.5. A standard Borel space is a measurable space isomorphic to [0 ,
1] equipped withits Borel σ -algebra. We call Borel the maps between two standard Borel spaces which are measurable.Let us give some notations regarding probability spaces : • If X is a measurable space, we denote by P ( X ) the set of probability measures on X . • If (
X, µ ) is a probability space and P is a property, we write ∀ ∗ x ∈ X P ( x ) for µ ( { x ∈ X : P ( x ) } ) = 1 and ∃ ∗ x ∈ X P ( x ) for µ ( { x ∈ X : P ( x ) } ) > • If (
X, µ ) is a probability space, Y is a measurable space and T : X → Y is a measurablemap, we write T ∗ µ for the pushforward of µ by T , that is the measure in P ( Y ) defined by T ∗ µ ( A ) = µ ( T − ( A )) for any Borel subset A ⊆ Y . Definition 2.6.
Let X be a standard Borel space and R be a Borel (as a subset of the measurablespace X × X ) equivalence relation on X . We let [ R ] be the group of Borel automorphisms of X whosegraphs are contained in R . We say that a Borel probability measure µ on X is R -invariant if everyelement of [ R ] preserves the measure µ , namely, ∀ T ∈ [ R ] , T ∗ µ = µ . Proposition 2.7 ([KM04, Section 8]) . With the same notations as above, for any µ ∈ P ( X ) , we candefine two measures µ l and µ r on R by • for all non-negative Borel f : R → [0 , ∞ ] , R R f dµ l = R X P y ∈ [ x ] R f ( x, y ) dµ ( x ) , • for all non-negative Borel f : R → [0 , ∞ ] , R R f dµ r = R X P y ∈ [ x ] R f ( y, x ) dµ ( x ) , where [ x ] R denotes the equivalence class of x for R . Then µ l = µ r if and only if µ is R -invariant. Definition 2.8.
Let G be a Borel graph on a standard probability space ( X, µ ) which has countableconnected components. Then the equivalence relation R G induced by G is the equivalence relation on( X, µ ) whose classes are the connected components of G . By the Lusin-Novikov theorem, R G is a Borelequivalence relation. We say that G is a graphing when µ is R G -invariant.We can define a measure on the set of edges of a graphing by: Definition 2.9.
Let G ( X, µ ) be a graphing and Z ⊆ E( G ) be a Borel set. The edge measure of theset Z is defined by µ E ( Z ) := µ l ( Z ) = µ r ( Z ), where µ l and µ r are defined with respect to the Borelequivalence relation R G .For a graphing of degree bound d , the edge measure of a set of edges is bounded by the measure ofthe vertices incident to this set. Namely, for all Borel Z ⊆ E( G ) we have12 µ (V inc ( Z )) ≤ µ E ( Z ) ≤ dµ (V inc ( Z )) . THE GENERALIZATION OF ROKHLIN LEMMA A measure-preserving transformation is called aperiodic if almost all its orbits are infinite.Rokhlin Lemma states that if T is an aperiodic measure-preserving transformation of a standardprobability space ( X, µ ), then for every n ∈ N and every ε >
0, there is a Borel subset A ⊆ X suchthat the sets A, T A, . . . , T n − A are pairwise disjoint and µ n − G i =0 T i A ! > − ε. What we present in this paper is not a generalization of Rokhlin Lemma itself but rather of one ofits important and well-known consequences:
Corollary 2.10 (Uniform Approximation Theorem, [Kec10, Theorem 2.2]) . Any two aperiodic measure-preserving transformations τ and τ on standard probability spaces ( X, µ ) and ( Y, ν ) are approximatelyconjugate. An aperiodic measure-preserving transformation can be seen as a free action of Z . The goal of thissection is to generalize the latter Corollary to hyperfinite actions of a countable group which have agiven IRS (i.e. Invariant Random Subgroup, defined in subsection 2.4). The key point on the proof of Uniform Approximation Theorem 2.10 is that the dynamics of an ape-riodic automorphism are understood on arbitrary large sets. In the section we define the notion ofhyperfiniteness of a pmp action, which allows one to make this idea work in a much more generalcontext.
Definition 2.11 (See "approximately finite group" in [Dye59]) . A pmp action Γ y ( X, µ ) is said tobe hyperfinite if for every finite S ⊆ Γ and every ε >
0, there exists a finite group G acting in ameasure-preserving way on ( X, µ ) such that µ ( { x ∈ X : S · x ⊆ G · x } ) > − ε. What we are mostly interested in is the characterization of hyperfiniteness for graphings.
Definition 2.12.
Let G ( X, µ ) be a graphing. G is called hyperfinite if for any ε > M ∈ N and a Borel set Z ⊆ E( G ) such that µ E ( Z ) < ε and the subgraphing H = G \ Z has componentsof size at most M . Definition 2.13.
Let F be a finite set. An F -colored graphing on a standard probability space( X, µ ) is a graphing G ( X, µ ) endowed with a Borel map ϕ G : E( G ) → F . For ( x, y ) ∈ E( G ), we call ϕ G ( x, y ) the color of ( x, y ).Additionally, for c ∈ F , we write E c ( G ) for the set of edges colored by c , namely ϕ − G ( c ).We will simply write G and consider the color implicitly when dealing with colored graphings. Definition 2.14.
Let G ( X, µ ) and G ′ ( Y, ν ) be two F -colored graphings. A colored graphing factormap π : Y → X is a pmp map such that for almost all y ∈ Y , π ↾ [ y ] G ′ is an isomorphism of F - coloredgraphs.We say that G is a colored factor of G ′ and we write G ⊑ c G ′ if there is a colored factor map π : Y → X .Let Γ be a group and S be a finite subset of Γ. Let us consider a measure-preserving actionΓ α y ( X, µ ). We define a P ( S )-colored graphing G α,S on ( X, µ ) by ( x, y ) ∈ E( G α,S ) if and only if thereis a s ∈ S such that y = sx and we color the edges of G α,S by letting the color of an edge ( x, y ) be { s ∈ S : y = sx } . We call it the Schreier graph of the action α relative to S . THE GENERALIZATION OF ROKHLIN LEMMA Lemma 2.15.
Let Γ be a countable group and let Γ α y ( X, µ ) be a pmp action. Then α is hyperfiniteif and only if for every finite S ⊆ Γ , G α,S is hyperfinite.Proof. Suppose α is hyperfinite and let S ⊆ Γ be finite and ε > G along with a pmp action G y ( X, µ ) such that µ ( { x ∈ X : S · x ⊆ G · x } ) > − ε . In particular, when restricted to the set { x ∈ X : S · x ⊆ G · x } ,the Schreier graph G α,S has finite components of size less than | G | .For the converse, suppose that for any S ⊆ Γ finite, the graphing G α,S is hyperfinite.Let S ⊆ Γ be finite and let ε >
0. Then there exist Z ⊆ E( G α,S ) Borel and M ∈ N such that µ E ( Z ) < ε and G α,S \ Z has components of size at most M .We define a pmp action of Q n ≤ M Z /n Z on ( X, µ ) as follows :Since (
X, µ ) is a standard probability space, there is a Borel linear ordering < of X . This induces,for n ≤ M , an action of Z /n Z on the set of elements of G α,S \ Z whose component is of size n byshifting any component according to the order < .It follows that Q n ≤ M Z /n Z acts as a product on X \ Z in a pmp way, and we extend this action tothe whole X by letting Q n ≤ M Z /n Z act trivially on Z .One can easily check that for x / ∈ V inc ( Z ), S · x is exactly the set of neighbors of x in G α,S \ Z and thus it is contained in [ x ] G α,S \ Z = (cid:16)Q n ≤ M Z /n Z (cid:17) · x . Moreover, µ (V inc ( Z )) ≤ µ E ( Z ) < ε so weconclude that α is hyperfinite. Let Γ α y ( X, µ ) be a measure-preserving action of the countable group Γ. With this action we canassociate a probability measure on the Polish space of subgroups of Γ as follows. Consider the compactPolish space { , } Γ . We let Sub(Γ) be the closed subset of { , } Γ consisting of the subgroups of Γ.Then Sub(Γ) is a compact Polish space.We have a natural map Stab α : X → Sub(Γ) defined by x Stab α ( x ) = { g ∈ Γ : g α ( x ) = x } andthat gives us a probability measure Stab α ∗ µ ∈ P (Sub(Γ)) that we call the Invariant Random Subgroup(IRS in short) of α and denote by θ α . Moreover, Γ acts on Sub(Γ) by conjugacy and the well knownformula Stab α ( gx ) = g Stab α ( x ) g − implies that the map Stab α is equivariant. Therefore, θ α is aΓ-invariant measure on Sub(Γ). We thus define the general notion of an IRS on Γ to be a probabilitymeasure on Sub(Γ) invariant for the action Γ y Sub(Γ) by conjugacy.G. Elek proved in [Ele12, Thm. 2] that two pmp actions of a countable group Γ with the same IRSare either both hyperfinite or both non-hyperfinite.Moreover, Abert, Glasner and Virag proved in [AGV14, Prop. 13] that any IRS can be obtainedas the IRS associated to a pmp action.We can thus express hyperfiniteness as a property of the IRS itself:
Definition 2.16.
Let Γ be a countable group. An IRS θ on Γ is called hyperfinite if one of thefollowing two equivalent statements is satisfied :1. There exists a hyperfinite pmp action which has IRS θ .2. Every pmp action which has IRS θ is hyperfinite. Definition 2.17.
Let Γ α y ( X, µ ) and Γ beta y ( Y, ν ). An action factor map π : Y → X is a measure-preserving map such that ∀ ∗ y ∈ Y ∀ γ ∈ Γ , π ( γ β y ) = γ α π ( y ).We say that α is a factor of β and we write α ⊑ β if there exists an action factor map π : Y → X . Lemma 2.18.
Let α, β be two actions of a countable group Γ on standard probability spaces ( X, µ ) and ( Y, ν ) . Suppose that there is an action factor map π : Y → X for α and β and that θ α = θ β . Then ∀ ∗ y ∈ Y, Stab α ( π ( y )) = Stab β ( y ) . THE GENERALIZATION OF ROKHLIN LEMMA Proof.
For γ ∈ Γ, let N γ = { Λ ∈ Sub(Γ) : γ ∈ Λ } . Then (N γ ) γ ∈ Γ is a subbasis of the topology ofSub(Γ) consisting of clopen sets and any measure on Sub(Γ) is determined by the values it takes onthis subbasis.By the definition of action factor map, we have ∀ ∗ y Stab β ( y ) ⊆ Stab α ( π ( y )). Suppose now that ∃ ∗ y Stab β ( y ) ( Stab α ( π ( y )).By countability of Γ, ∃ γ ∈ Γ ∃ ∗ y , γ ∈ Stab α ( π ( y )) \ Stab β ( y ), thus θ β (N γ ) = Stab β ∗ ν (N γ ) < (Stab α ◦ π ) ∗ ν (N γ )= Stab α ∗ ( π ∗ ν )(N γ )= Stab α ∗ µ (N γ )= θ α (N γ ) , a contradiction. Corollary 2.19.
Let α, β be actions of a countable group Γ on standard probability spaces ( X, µ ) and ( Y, ν ) such that α ⊑ β and θ α = θ β , and let S ⊆ Γ be finite . Then we have G α,S ⊑ c G β,S as P ( S ) -coloredgraphings.Proof. Applying Lemma 2.18 to an action factor map π : Y → X gives us that for almost every y ∈ Y , π ↾ Γ · y is a Γ-equivariant bijection Γ · y → Γ · π ( y ) and so it is an isomorphism of Schreier graphs. Itfollows that π is a graphing factor map. We begin with the case where one of the actions is a factor of the other. In fact we prove a strongerversion involving the stability of Borel sets.
Definition 2.20.
Let F , F be two finite sets. An ( F , F ) -bicolored graphing on a standardprobability space ( X, µ ) is a graphing G ( X, µ ) endowed with two Borel maps ϕ G : E( G ) → F and ψ G : X → F . We call ψ G ( x ) the vertex-color of x and ϕ G ( x, y ) the edge-color of ( x, y ). Definition 2.21.
Let G ( X, µ ) and G ′ ( Y, ν ) be two ( F , F )-bicolored graphings. A bicolored graph-ing factor map π : Y → X is an F -colored graphing factor map such that ψ G ◦ π = ψ G ′ .We say that G is a bicolored factor of G ′ and we write G ⊑ bic G ′ if there is a bicolored factor map π : Y → X . Theorem 2.22 (Approximate parametrized conjugacy for factor actions) . Let ( X, µ ) and ( Y, ν ) bestandard probability spaces and A , . . . , A k ⊆ X , B , . . . , B k ⊆ Y be Borel subsets. Let Γ be a countablegroup, θ be a hyperfinite IRS on Γ and Γ α y ( X, µ ) , Γ β y ( Y, ν ) be pmp actions of Γ with IRS θ andsuch that α ⊑ β for an action factor map π : Y → X such that ∀ i ≤ k, π − ( A i ) = B i . Then for ε > and γ , . . . , γ n ∈ Γ , there exists a pmp bijection ρ : X → Y such that ∀ i ≤ k, ρ ( A i ) = B i and µ ( { x ∈ X : ∀ i ≤ n, ρ ◦ γ αi ( x ) = γ βi ◦ ρ ( x ) } ) > − ε. Proof.
We begin the proof with a claim about graphings.
Claim 2.22.1.
Let G ( X, µ ) and G ′ ( Y, ν ) be hyperfinite ( F , F )-bicolored graphings of degreebound at most d such that G ( X, µ ) ⊑ bic G ′ ( Y, ν ). Then for any ε > ρ : X → Y such that ψ G = ψ G ′ ◦ ρ and µ E [ c ∈ F ρ − (cid:0) E c ( G ′ ) (cid:1) △ E c ( G ) < ε. THE GENERALIZATION OF ROKHLIN LEMMA Proof.
Let π be a bicolored graphing factor map Y → X . First take a Borel set Z ⊆ E( G ) ofmeasure less than ε d and M ∈ N such that the graphing H = G \ Z has components of size atmost M . Let Z ′ = π − ( Z ) and H ′ = G ′ \ Z ′ . Since π is a graphing factor map, we know that H ′ has components of size at most M . Then H and H ′ have a ( F , F )-bicolored graphing structurerespectively for the maps ϕ G ↾ E( H ) , ψ G and ϕ G ′ ↾ E( H ′ ) , ψ G ′ .Consider the set G M of connected ( F , F )-colored graphs of size at most M . We consider thetwo partitions X = F S ∈ G M C H S and Y = F S ∈ G M C H ′ S , where C H S is defined to be the set of vertices of H whose component is ( F , F )-colored isomorphic to S . Since π induces ( F , F )- colored graphisomorphisms, we have C H ′ S = π − ( C H S ).In order to define ρ , it suffices to define a measure-preserving bijection ρ S : C H S → C H ′ S preservingbicolored graph structures for each S ∈ G M .Indeed, the union of all these bijections would yield a measure-preserving bijection ρ : X → Y preserving vertex-colors such that ∀ x ∈ X \ V inc ( Z ) , B G ( x,
1) = B H ( x, ≃ B H ′ ( ρ ( x ) ,
1) =B G ′ ( ρ ( x ) , G ( v, n ) denotes the ball of size n centered at v in the graph G . Hence wewould have V inc S c ∈ F ρ − (E c ( G ′ )) △ E c ( G ) ! ⊆ V inc ( Z ), and so µ E [ c ∈ F ρ − (cid:0) E c ( G ′ ) (cid:1) △ E c ( G ) ≤ dµ (V inc ( Z )) ≤ dµ E ( Z ) < ε. Take S ∈ G M and let us define ρ S . First we define a partition of C H S into Borel transversals( T v ) v ∈ V( S ) (for H ) by induction, such that the elements of T v occupy the same place in theircomponent for H as v in S .Suppose that the T v ′ are already defined for v ′ ∈ R where R is a proper subset of V( S ). Take v ∈ V( S ) \ R incident to R and let f T v = { x ∈ C H S : ([ x ] H , x ) ≃ R ( S, v ) } . Here ≃ R meansisomorphic over R , that is there exists an isomorphism f : ([ x ] H , x ) → ( S, v ) of colored rootedgraphs such that ∀ v ′ ∈ R, f ([ x ] H ∩ T v ′ ) = { v ′ } . Now since H has finite components, chose for T v any Borel transversal of f T v . Then we let R ′ = R ∪ { v } and we iterate the construction.Again since π is a bicolored graphing factor map, the family ( π − ( T v )) v ∈ V( S ) is a partition of C H ′ S into Borel transversals (for H ′ ) such that the elements of π − ( T v ) occupy the same place intheir component for H ′ as v in S . We may now define ρ S : – We start by chosing v ∈ S and taking a measure-preserving bijection ρ v S : T v → π − ( T v ). – Then for every v ∈ S , there is a unique way of extending ρ v S to T v while respecting thegraph structure of S . Indeed, take x ∈ T v , there is a unique x ∈ [ x ] H ∩ T v and we wantto define ρ vS ( x ) ∈ [ ρ v S ( x )] H ′ ∩ π − ( T v ) but again this intersection is a singleton. Define ρ S : C H S → C H ′ S to be this unique extension of ρ v S satisfying the condition above.As π is a colored graphing factor map, it is clear that ρ S is a measure-preserving bijectionand that for every x ∈ C H S , ρ S induces an isomorphism of colored graphs between [ x ] H and[ ρ S ( x )] H ′ . (cid:4) We now want to apply the Claim to suitable graphings to conclude. Let S = { γ , . . . , γ n , γ − , . . . , γ − n } and consider the graphings G α,S and G β,S .For the spaces of colors, we choose F = P ( S ) and F = P ( { , . . . , k } ). The way we color edgeshas already been explained; for vertices, simply color a vertex x ∈ X by ψ G α,S ( x ) = { i ≤ k : x ∈ A i } and y ∈ Y by ψ G β,S ( y ) = { i ≤ k : y ∈ B i } .First, G α,S and G β,S are indeed ( P ( S ) , P ( { , . . . , k } ))-bicolored graphings, and are hyperfinitesince α and β are hyperfinite actions.The next step is to prove that π considered in the statement of the theorem is a bicolored factormap for the ( P ( S ) , P ( { , . . . , k } ))-bicolored graphings G α,S and G β,S . THE GENERALIZATION OF ROKHLIN LEMMA • First, π is indeed a pmp map Y → X . • Then for y ∈ Y , we have ψ G α,S ( π ( y )) = { i ≤ k : π ( y ) ∈ A i } = { i ≤ k : y ∈ B i } = ψ G β,S ( y ) . • Finally, by Corollary 2.19, π is furthermore a colored graphing factor map between the P ( S )-colored graphings G α,S and G β,S .Applying the Claim gives us a pmp bijection ρ : X → Y such that ψ G α,S = ψ G β,S ◦ ρ and µ E [ c ∈ P ( S ) E c ( G α,S ) △ ρ − (E c ( G β,S )) < ε . But then for 1 ≤ i ≤ k , ρ ( A i ) = B i , and by definitions of G α,S and G β,S we get { x ∈ X : ∃ γ ∈ S, ρ ◦ γ α ( x ) = γ β ◦ ρ ( x ) } ⊆ V inc [ c ∈ P ( S ) E c ( G α,S ) △ ρ − (E c ( G β,S )) , so its measure is less than 2 · ε = ε . To conclude the proof of Theorem A, we will use the transitivity of the approximate conjugacy relationand show that for any two pmp actions Γ α y ( X, µ ) and Γ β y ( Y, ν ) of Γ such that θ α = θ β , there is athird pmp action Γ ζ y ( Z, η ) of IRS θ such that both α and β are factors of ζ .We recall the definition of the relative independent joining following the presentation in [Gla03]. Proposition 2.23 (Disintegration theorem,[Gla03, A.7]) . Let
X, Y be standard probability spaces, µ ∈ P ( Y ) and π : Y → X be a measurable map. We let ν = π ∗ µ . Then there is a ν -a.e. uniquelydetermined family of probability measures ( µ x ) x ∈ X ∈ P ( Y ) X such that:1. For each Borel B ⊆ Y , the map x µ x ( B ) is measurable.2. For ν -a.e. x ∈ X , µ x is concentrated on the fiber π − ( x ) .3. For every Borel map f : Y → [0 , ∞ ] , R Y f ( y ) dµ ( y ) = R X R Y f ( y ) dµ x ( y ) dν ( x ) .We then write µ = R X µ x dν . Definition 2.24 ([Gla03, Section 6.1]) . Let Γ α y ( X, µ ) and Γ β y ( X ′ , µ ′ ) be pmp actions on standardprobability spaces, and let Γ ξ y ( Y, ν ) be an action on a standard probability space common factor of α and β for respective action factor maps π : X → Y and π ′ : X ′ → Y .We can disintegrate µ and µ ′ with respect to ν using the Borel maps π and π ′ to get µ = R Y µ y dν and µ ′ = R Y µ ′ y dν .Consider Z := X × Y and η ∈ P ( Z ) defined by η = R Y µ y × µ ′ y dν .The pmp action Γ α × β y ( Z, η ) is called the independent joining of α and β over ξ and is denotedby α × ξ β .The action α × ξ β is indeed a joining of α and β over ξ , meaning that both α and β are factors oftheir independent joining over ξ , respectively for the projections on the first and second coordinates p and p , and moreover the following diagram commutes, up to a null set: MODEL THEORY OF HYPERFINITE ACTIONS α × ξ βα βξ p p π π Let θ be an IRS on Γ, we write θ for the measure-preserving conjugation action Γ θ y (Sub(Γ) , θ ).For every pmp action Γ α y ( X, µ ), the map Stab α : ( X, µ ) → (Sub(Γ) , θ ) is an action factor map. Lemma 2.25.
Let Γ be a countable group and θ be an IRS on Γ . Let Γ α y ( X, µ ) , Γ β y ( Y, ν ) be pmpactions of IRS θ . Then α × θ β has IRS θ .Proof. Let ζ denote α × θ β . We know that the following diagram commutes. ζα β θ p p Stab α Stab β Therefore, for γ ∈ Γ, we have ∀ ∗ ( x, y ) , γx = x ⇔ γy = y ⇔ γ ( x, y ) = ( x, y ). It follows that ∀ ∗ ( x, y ) , Stab ζ ( x, y ) = Stab α ( x ) or in other words, Stab ζ = Stab α ◦ p . We conclude that θ ζ = Stab ζ ∗ η = Stab α ∗ ( p ∗ η ) = Stab α ∗ µ = θ α = θ. Theorem A states that if α and β are two pmp hyperfinite actions of a group Γ on a standardprobability space such that θ α = θ β , then α and β are approximately conjugate. We can now provethis theorem: Proof.
Let Γ α y ( X, µ ) and Γ β y ( Y, ν ) be two hyperfinite actions of Γ having IRS θ and consider thejoining Γ ζ y ( Z, η ) from Lemma 2.25.Applying twice Theorem 2.22 with no Borel parameters we get two pmp bijections ρ : X → Z and ρ ′ : Y → Z such that: µ ( { x ∈ X : ∀ i ≤ n, ρ ◦ γ αi ( x ) = γ ζi ◦ ρ ( x ) } ) > − ε ν ( { y ∈ Y : ∀ i ≤ n, ρ ′ ◦ γ βi ( y ) = γ ζi ◦ ρ ′ ( y ) } ) > − ε . Thus, ρ ′− ◦ ρ : X → Y witnesses the ε -approximate conjugacy of α and β . The reader unfamiliar with continuous model theory is referred to [BYBHU08]. We will use the samenotations as theirs.
Definition 3.1. A measure algebra is a Boolean algebra ( A , ∨ , ∧ , ¬ , , , ⊆ , △ ) endowed with afunction µ : A → [0 ,
1] satisfying the following :
MODEL THEORY OF HYPERFINITE ACTIONS µ (1) = 1.2. ∀ a, b ∈ A , µ ( a ∧ b ) = 0 ⇒ µ ( a ∨ b ) = µ ( a ) + µ ( b ).3. The function d µ ( a, b ) := µ ( a △ b ) is a complete metric on A . Proposition 3.2 ([Fre02, 323G c)]) . Any measure algebra A is Dedekind complete, meaning that anysubset S ⊆ A admits a supremum and an infimum, that we respectively denote by W S and V S . Definition 3.3.
An element a ∈ A is an atom if ∀ b ∈ A , b ⊆ a ⇒ b ∈ { , a } . A measure algebra is atomless if it has no atoms. Proposition 3.4 ([Fre02, 331C]) . If a measure algebra A is atomless, then ∀ a ∈ A ∀ r ∈ [0 , µ ( a )] ∃ b ⊆ a, µ ( b ) = r. We introduce the classical example of a measure algebra: For (
X, µ ) a probability space, we letMAlg(
X, µ ) be the quotient of the Boolean algebra of measurable subsets of X by the σ -ideal of nullsets. For A ⊆ X Borel we denote its class in MAlg(
X, µ ) by [ A ] µ . The measure µ descends to thequotient MAlg( X, µ ) and then MAlg(
X, µ ) endowed with µ is a measure algebra. When ( X, µ ) is astandard probability space, MAlg(
X, µ ) is atomless and separable for the topology induced by d µ .Conversely, we have: Proposition 3.5 ([Fre02, 331L]) . Let A be a separable atomless measure algebra. Then there exists astandard probability space ( X, µ ) such that A is isomorphic to MAlg(
X, µ ) . Let f : ( X, µ ) → ( Y, ν ) be a measure-preserving map. Then the map e f : MAlg( Y, ν ) → MAlg(
X, ν )sending [ A ] ν to (cid:2) f − ( A ) (cid:3) µ is a measure algebra morphism. Moreover, if f is a bimeasurable bijection,then e f is an isomorphism.However, in general, given a morphism ϕ : MAlg( X, ν ) → MAlg(
Y, µ ) there is no way to get a liftingof ϕ , that is a point to point measure-preserving map ϕ : Y → X such that e ϕ = ϕ . However, in thecase of standard probability spaces, such a construction exists: Proposition 3.6 ([Fre13, 425D]) . Let ( X, µ ) and ( Y, ν ) be standard probability spaces. For every mor-phism of measure algebras ϕ : MAlg( X, µ ) → MAlg(
Y, ν ) there is a lifting ϕ : Y → X of ϕ . Moreover,for Γ a countable group acting by automorphisms on MAlg(
X, µ ) by an action α , there is a lifting of α ,that is an action Γ α y X acting by measure-preserving transformations such that ∀ γ ∈ Γ , f γ α = (cid:0) γ − (cid:1) α . We axiomatize the theory AMA of atomless measure algebras in the signature L = {∨ , ∧ , ¬ , , } ( △ is defined as usual) as in [BYBHU08, Section 16]. Proposition 3.7 ([BYBHU08, 16.2]) . The theory
AMA is separably categorical and therefore complete.
We also have:
Proposition 3.8 ([BYBHU08, 16.6 and 16.7]) . The theory
AMA admits quantifier elimination. More-over, the definable closure dcl M ( C ) of a subset C in a model M of AMA is the substructure h C i of M generated by C . We will now give a characterization of the types in the theory AMA. For that we need a little bitof terminology.To any measure algebra A we can associate a natural Hilbert space L ( A ) called the L space of A . This construction is consistent in the sense that if A is the measure algebra of a probability space( X, µ ), then there is a natural linear isometry between L ( A ) and L ( X, µ ). MODEL THEORY OF HYPERFINITE ACTIONS Definition 3.9.
Let A be a measure algebra and B a measure subalgebra of A . Then the space L ( B )is a closed vector subspace of the Hilbert space L ( A ), we denote by P B the orthogonal projection onL ( B ) and we call it the conditional expectation with respect to B . Particularly, for a ∈ A , a canbe seen as the element a of L ( A ) and we call P B ( a ) the conditional probability of a with respectto B . For simplicity, we will denote it by P B ( a ).By definition, the conditional probability of a with respect to B is the only B -measurable functionsuch that for any B -measurable function f , we have R P B ( a ) f = R a f . Proposition 3.10 ([BYBHU08, 16.5]) . Let M | = AMA , ¯ a, ¯ b be n -uples of elements of M and C ⊆ M .Then tp(¯ a/C ) = tp(¯ b/C ) if and only if for every map σ : { , . . . , n } → {− , } we have P h C i ^ ≤ i ≤ n a σ ( i ) i = P h C i ^ ≤ i ≤ n b σ ( i ) i , where a denotes a and a − denotes its complement ¬ a in M . A θ Until now, we studied actions of any countable group. Since any action of a countable group canbe represented as an F ∞ -action, for the sake of simplicity, we now restrict to F ∞ -actions, where F ∞ denotes the countably generated free group.We now expand the signature L with a countable set of function symbols indexed by F ∞ , that weidendify with F ∞ itself. We call this new signature L ∞ . We begin by considering the theory A F ∞ consisting of the following axioms: • The axioms of AMA. • For γ ∈ F ∞ , the axioms expressing that γ is a measure algebra isomorphism: – sup a,b d ( γ ( a ∨ b ) , γa ∨ γb ) = 0 – sup a,b d ( γ ( a ∧ b ) , γa ∧ γb ) = 0 – sup a | µ ( γa ) − µ ( a ) | = 0 – sup a inf b d ( a, γb ) = 0 • The axioms expressing that F ∞ acts on the measure algebra: – sup a d (1 F ∞ a, a ) = 0 – For γ , γ ∈ F ∞ , the axiom sup a d ( γ ( γ a ) , ( γ γ ) a ) = 0By Propositions 3.5 and 3.6 any separable model of A F ∞ can be seen as the action on a measurealgebra associated with a measure-preserving action F ∞ y ( X, µ ) on a standard probability space. If α is a pmp action on a probability space, we write M α for the model of A F ∞ induced by α . Withoutloss of generality, from now on, separable models we consider are always of the form M α for α a pmpaction on a standard probability space. Definition 3.11.
For f any measure-preserving transformation ( X, µ ) → ( X, µ ), where (
X, µ ) is aprobability space, we call the set { x ∈ X : f x = x } the support of f and we denote it by Supp f . Definition 3.12.
Let ( A , µ ) be a measure algebra, the support of an automorphism ϕ of A is definedby supp ϕ = V { a ∈ A : ∀ b ⊆ ¬ a, ϕb = b } .It is classic that if f is a measure-preserving transformation of a standard probability space ( X, µ ),then [Supp f ] µ = supp e f .Our goal is now to give a first order description of the support of an automorphism of a separablemeasure algebra: MODEL THEORY OF HYPERFINITE ACTIONS Lemma 3.13.
1. Let ϕ be an automorphism of a separable atomless measure algebra A such that supp ϕ = 0 . Then there exists b = 0 ∈ A such that ϕb ∧ b = 0 .2. Let A be a separable atomless measure algebra. Let ϕ be an automorphism of A .Then there is a ∈ A such that supp ϕ = ϕ − a ∨ a ∨ ϕa and a ∧ ϕa = 0 . Furthermore, wehave supp ϕ = W { ϕ − a ∨ a ∨ ϕa : a ∈ A , a ∧ ϕa = 0 } .Proof.
1. Consider a standard probability space (
X, µ ) such that MAlg(
X, µ ) = A and let f be aBorel lifting of ϕ to X . Since X is standard, let ( B n : n ∈ N ) be a countable family of Borel subsetsof X separating the points. Without loss of generality, we may suppose that the set B n : n ∈ N is stable by the operation of complement. For n ∈ N , let B ′ n = B n \ f − ( B n ). For x ∈ Supp f ,there is n such that x ∈ B n and f ( x ) / ∈ B n so x ∈ B ′ n and therefore µ ( S n ∈ N B ′ n ) ≥ µ (Supp f ) > n such that B ′ n is of positive measure and let b = [ B ′ n ] µ .2. First A is a measure algebra and therefore is complete as a Boolean algebra so it has a maximalelement a disjoint from its image by ϕ .Consider b = ϕ a \ ( ϕ − a ∨ a ∨ ϕa ). We have( a ∨ b ) ∧ ϕ ( a ∨ b ) = ( a ∧ ϕa ) ∨ ( a ∧ ϕb ) ∨ ( b ∧ ϕa ) ∨ ( b ∧ ϕb ) ⊆ ∨ ( a \ a ) ∨ ( ϕa \ ϕa ∨ ( ϕ a \ ϕ a )= 0 . Thus a ∨ b is disjoint from its image. By maximality of a , we then have b ⊆ a , but by definition b ∧ a = 0, so b = 0, or in other words, ϕ a ⊆ ϕ − a ∨ a ∨ ϕa .It follows that ϕ (cid:0) ϕ − a ∨ a ∨ ϕa (cid:1) ⊆ ϕ − a ∨ a ∨ ϕa and since ϕ preserves the measure, theset ϕ − a ∨ a ∨ ϕa is invariant by ϕ .Furthermore, a is disjoint from its image by ϕ , and so ϕ − a and ϕa are also disjoint fromtheir respective image, so we have ϕ − a ∨ a ∨ ϕa ⊆ supp ϕ. Conversely, let c = supp ϕ \ ( ϕ − a ∨ a ∨ ϕa ) and suppose that c = 0. Since c is invariant by ϕ , we can consider the automorphism ϕ ↾ c of the measure algebra lying under c . Applying thefirst point of this lemma to this automorphism, we get a non trivial b ⊆ c disjoint from its imageby ϕ .But then, a ∨ b contradicts the maximality of a . We conclude that ϕ − a ∨ a ∨ ϕa = supp ϕ. Finally, as we already noticed, any set of the form ϕ − a ∨ a ∨ ϕa for a ∧ ϕa = 0 is a subset ofsupp ϕ , so we have supp ϕ = _ { ϕ − a ∨ a ∨ ϕa : a ∈ A , a ∧ ϕa = 0 } . Now we can prove that the IRS of a pmp action on a measure algebra is determined by the theoryof this action seen as a model of A F ∞ . Definition 3.14.
For γ ∈ F ∞ we let t γ ( a ) denote the term γ − ( a \ γa ) ∨ ( a \ γa ) ∨ γ ( a \ γa ).It follows from Lemma 3.13 that for M | = A F ∞ , supp γ = W { t γ ( a ) : a ∈ M } . Lemma 3.15.
Let γ ∈ F ∞ . Then the support of γ is definable without parameters in the theory A F ∞ . MODEL THEORY OF HYPERFINITE ACTIONS Proof.
We need to prove that the distance to supp γ is definable. By definition of the distance, wehave ∀ a ∈ M , d ( a, supp γ ) = µ ( a \ supp γ ) + µ (supp γ \ a ).On the one hand, µ ( a \ supp γ ) = inf b µ ( a \ t γ ( b )) so the first part is definable.On the other hand, µ (supp γ \ a ) = sup b µ ( t γ ( b ) \ a ) and therefore the second part is definable aswell. Theorem 3.16.
Let M α , M β be two elementarily equivalent models of A F ∞ . Then θ α = θ β .Proof. As θ α and θ β are measures on Sub( F ∞ ), they are determined by their values on the setsN F,G = { Λ ≤ F ∞ : F ⊆ Λ , G ∩ Λ = ∅ } where F and G are finite.Note that θ α (N F, ∅ ) = µ ( T γ ∈ F Supp γ α ) and θ β (N F, ∅ ) = µ ( T γ ∈ F Supp γ β ), but by Lemma 3.13 thesesupports are the same as those defined in the measure algebra. Furthermore, by Lemma 3.15, for each γ ∈ F ∞ , supp γ is definable over ∅ in the theory F ∞ , and since the definable closure is a substructure,then V γ ∈ F supp γ must be definable over ∅ as well. Thus by elementary equivalence, for every finite F ⊆ F ∞ , we have θ α (N F, ∅ ) = θ β (N F, ∅ ).Now for F, G finite subsets of F ∞ , write N F,G = N F, ∅ \ S γ ∈ G N F ∪{ γ } , ∅ . By the inclusion-exclusionprinciple, we then get θ α (N F,G ) = θ α (N F, ∅ ) + | G | X i =1 ( − i X { J ⊆ G : | J | = i } θ α (N F ∪ J, ∅ )= θ β (N F, ∅ ) + | G | X i =1 ( − i X { J ⊆ G : | J | = i } θ β (N F ∪ J, ∅ )= θ β (N F,G ) . For θ an IRS, let A θ be the L ∞ -theory consisting of: • The axioms of A F ∞ . • For F ⊆ F ∞ finite, the axiom sup { a γ : γ ∈ F } µ ( V γ ∈ F t γ ( a γ )) = θ (N F, ∅ ).Then the models of A θ are exactly the measure-preserving actions of F ∞ which have IRS θ . Definition 3.17.
Let (
X, µ ) be a standard probability space and Γ be a countable group.First, let Aut(
X, µ ) be the space of automorphisms of MAlg(
X, µ ). We equip it with a completemetric d u called the uniform metric and defined by the formula d u ( ϕ, ψ ) := sup a ∈ MAlg(
X,µ ) d µ ( ϕa, ψa ).We call the topology induced the uniform topology .Then we define the space A (Γ , X, µ ) of pmp actions of Γ on ( X, µ ) naturally as a subspace ofAut(
X, µ ) Γ . The uniform topology on Aut( X, µ ) gives rise to a product topology on Aut(
X, µ ) Γ whichis completely metrizable and for which A (Γ , X, µ ) is closed. Again, we call this topology the uniformtopology on A (Γ , X, µ ).From now on, fix a complete metric d u compatible with the uniform topology on A ( F ∞ , X, µ ). Theorem 3.18.
Let ϕ (¯ x, ¯ y ) be an L ∞ -formula, where | ¯ x | = n , | ¯ y | = m , let ( X, µ ) be a standardprobability space and let ¯ p ∈ MAlg(
X, µ ) m .Then the map ( A ( F ∞ , X, µ ) , d u ) −→ ( l ∞ (MAlg( X, µ ) n ) , k k ∞ ) α (cid:16) ϕ M α (¯ a, ¯ p ) (cid:17) ¯ a ∈ MAlg(
X,µ ) n is uniformly continuous. MODEL THEORY OF HYPERFINITE ACTIONS Proof.
We prove this result by induction on formulas. For now assume that the theorem holds foratomic formulas. First remark that if the theorem holds for certain formulas, then it holds for anycombination of these formulas constructed with the help of connectives, by using their uniform conti-nuity. Then it suffices to treat the case of quantifiers to conclude. But it is immediate, since we usethe norm k k ∞ .Let us now prove the theorem for atomic formulas. If ϕ (¯ x, ¯ y ) is an atomic formula, then it isequivalent to a formula of the form ϕ (¯ x, ¯ y ) := µ ( t ( γ ¯ x, . . . , γ l ¯ x, γ ¯ y, . . . , γ l ¯ y ) for an L -term t and some γ , . . . , γ l ∈ F ∞ . Let ε > δ > z and¯ z ′ ∈ MAlg(
X, µ ) ( n + m ) l , if d µ (¯ z, ¯ z ′ ) < δ then d µ ( t (¯ z ) , t ( ¯ z ′ )) < ε .Now if α, β ∈ A ( F ∞ , X, µ ) are sufficiently d u -close, then for every a ∈ MAlg(
X, µ ) and 1 ≤ i ≤ l , d µ ( γ αi a, γ βi a ) < δ . It follows that for all ¯ a ∈ MAlg(
X, µ ) n , (cid:12)(cid:12)(cid:12) ϕ M α (¯ a, ¯ p ) − ϕ M β (¯ a, ¯ p ) (cid:12)(cid:12)(cid:12) ≤ d µ (cid:16) t ( γ α ¯ a, . . . , γ αl ¯ a, γ α ¯ p, . . . , γ αl ¯ p ) , t ( γ β ¯ a, . . . , γ βl ¯ a, γ β ¯ p, . . . , γ βl ¯ p ) (cid:17) < ε, which finishes the proof. Theorem 3.19.
Let θ be a hyperfinite IRS on F ∞ . Then the theory A θ is model complete.Proof. It suffices to show that any inclusion of two separable models is elementary. Indeed, supposethis result and take any M ⊆ N | = A θ , ϕ (¯ x ) a L ∞ -formula and ¯ p ∈ M finite. By the Löwenheim-Skolem theorem, find a separable M ′ (cid:22) M containing ¯ p . Again by the Löwenheim-Skolem theorem,find a separable N ′ (cid:22) N containing the separable structure M ′ . Using the hypothesis, M ′ (cid:22) N ′ sowe finally get ϕ (¯ p ) M = ϕ (¯ p ) M ′ = ϕ (¯ p ) N ′ = ϕ (¯ p ) N . Let M ⊆ N be two separable models of A θ . Consider a L ∞ -formula ϕ (¯ x ) with k variables and¯ p ∈ MAlg(
X, µ ) k .A classical argument derived from Proposition 3.6 allows us to chose two pmp actions F ∞ α y ( X, µ )and F ∞ β y ( Y, ν ) on standard probability spaces along with a pmp map π : Y → X , such that M ≃ M α , N ≃ M β , and π is a lifting of the inclusion MAlg( X, µ ) ֒ → MAlg(
Y, ν ), which is equivariantrespectively to the actions α and β . For 1 ≤ i ≤ k , let A i ⊆ X be a Borel representative of p i and let B i = π − ( A i ), which is also a Borel representative of p i , in Y .Then by Theorem 2.22, α is in the uniform closure of the set C ( β ) := { ρ − βρ : ρ is a pmp bijection X → Y such that ∀ i ≤ k, ρ − ( A i ) = B i } . But then Theorem 3.18 implies that ϕ M α (¯ p ) ∈ { ϕ M β ′ (¯ p ) : β ′ ∈ C ( β ) } . Furthermore, for any β ′ ∈ C ( β ),we have ( β ′ , ¯ A ) ≃ ( β, ¯ B ), so that ( M β ′ , ¯ p ) ≡ ( M β , ¯ p ) and consequently ϕ M β ′ (¯ p ) = ϕ M β (¯ p ). Thisestablishes that ϕ M α (¯ p ) = ϕ M β (¯ p ).Hence M α (cid:22) M β and therefore A θ is model complete.Now for completeness we combine model completeness with the argument of amalgamation alreadyseen in Section 2.5.2. Theorem 3.20.
Let θ be a hyperfinite IRS on F ∞ . Then the theory A θ is complete.Proof. As usual, it is sufficient to prove that two separable models of A θ are elementarily equivalent.Let M α , M β | = A θ be two separable models and consider the action ζ := α × θ β . By Lemma 2.25,we have M ζ | = A θ and moreover, both M α and M β are substructures of M ζ .Now since A θ is model complete, we have M α (cid:22) M ζ and M β (cid:22) M ζ , so M α ≡ M ζ ≡ M β . MODEL THEORY OF HYPERFINITE ACTIONS Proposition 3.21 ([BYBHU08, Prop. 13.16]) . Let T be a countable theory. Then T admits quantifierelimination if and only if for any M , N | = T , any substructure A ⊆ M and any embedding f : Z ֒ → N ,there is an elementary extension N ′ of N and an embedding ˜ f : M ֒ → N ′ extending f . Definition 3.22.
We say that a theory T admits amalgamation if for any M , M | = T and anycommon substructure Z , there is a joining of M and M over Z , that is a structure N | = T andembeddings M i ֒ → N ( i = 1 ,
2) such that the following diagram commutes: NM M Z The next lemma is a classical result in discrete model theory and it easily extends to continuousmodel theory.
Lemma 3.23.
Let T be a theory. Then T admits quantifier elimination if and only if it admitsamalgamation and is model complete.Proof. Suppose that T admits quantifier elimination. Let M , M | = T with a common substructure Z , applying Proposition 3.21 where f is the inclusion Z ֒ → M , we get N as required.Now let M ⊆ N be two models of T . By quantifier elimination, we only need to prove that M | = ϕ (¯ a ) ⇔ N | = ϕ (¯ a ) for atomic formulas ϕ and finite tuples ¯ a of parameters in M . But this istrivial by the definition of inclusion for models.Conversely, suppose T admits amalgamation and is model complete and let M , N | = T , Z ⊆ M be a substructure, and f : Z ֒ → N . By considering a monster model, we may suppose that Z ⊆ N and f is the identity. Then by amalgamation there is a model N ′ | = T and embeddings ϕ, ψ such thatthe following diagram commutes: N ′ M NZ ϕ ψ
Id Id
Again we may suppose that N ⊆ N ′ and ψ is the identity, thus by model completeness we have N (cid:22) N ′ . Furthermore, the diagram now exactly states that ϕ extends the inclusion Z ֒ → N .In order to prove that our theories eliminate quantifiers, it only remains to prove that they haveamalgamation. However, the following example shows that this is not the case in general. Definition 3.24.
Let Γ α y X be an action of a group on a standard Borel space. We say that µ ∈ P ( X )is ergodic if every Γ-invariant for α measurable subset of X is either null or connull for µ .It can be shown that ergodic measures are the extreme points of the convex space P ( X ).For Invariant Random Subgroups, we consider the notion of ergodicity with respect to the actionΓ y Sub(Γ) by conjugation.
Proposition 3.25.
Let θ be a non-ergodic IRS on F ∞ . Then A θ does not have quantifier elimination. MODEL THEORY OF HYPERFINITE ACTIONS Proof.
Take any finite subset F ⊆ F ∞ . Then µ x ∧ V γ ∈ F supp γ ! := sup { a γ : γ ∈ F } µ x ∧ V γ ∈ F t γ ( a γ ) ! isa definable predicate in the signature L ∞ . However, as we shall see, not all predicates of this form aredefinable without quantifiers.Indeed, suppose that for every finite subset F ⊆ F ∞ , there is a quantifier free formula ϕ F ( x )equivalent to µ x ∧ V γ ∈ F supp γ ! .Write θ = tθ + (1 − t ) θ for a t ∈ (0 , ] and θ = θ two IRSs on F ∞ . Let κ be a pmp action on([0 , , λ ) with IRS θ and κ be a pmp action on ([0 , , λ ) with IRS θ . Define • F ∞ α y ( X = [0 , × { , , } , µ = tλ × δ + tλ × δ + (1 − t ) λ × δ ) that acts like κ on [0 , × { } and acts like κ both on [0 , × { } and on [0 , × { } . • F ∞ β y ( X = [0 , × { , , } , µ = tλ × δ + tλ × δ + (1 − t ) λ × δ ) that acts like κ on [0 , × { } and acts like κ both on [0 , × { } and on [0 , × { } .We have θ α = θ β = θ .Let M be the finite measure algebra generated by three atoms { a, b, c } of respective measure t , t and 1 − t . By sending a to [0 , × { } , b to [0 , × { } and c to [0 , × { } , one can embed M in both M α and M β . Then M endowed with the trivial action is a common substructure of M α and M β .As ϕ F ( x ) is quantifier free, we have ϕ M α F ( a ) = ϕ M F ( a ) = ϕ M β F ( a ), but M α | = µ ( a ∧ ^ γ ∈ F supp γ ) = tθ (N F ) whereas M β | = µ ( a ∧ ^ γ ∈ F supp γ ) = tθ (N F ) . Since an IRS is determined by its values on the sets of the form N F , we get θ = θ , a contradiction.Thus, non-ergodicity of the IRS is an obstacle to quantifier elimination. A natural question is toask about a converse: For which θ does the theory A θ admit quantifier elimination? Is it the case for any ergodic IRS? The author does not have any satisfying answer.However, we answer another interesting question. One can ask what we can reasonably add to thetheory A θ to expand it into a theory A ′ θ in a signature L ′∞ ⊇ L ∞ which has quantifier elimination.The issue encountered in Proposition 3.25 is that formulas involving the supports of the elements of F ∞ may not be equivalent to quantifiers free formulas in A θ . This motivates us to look at expansionsthat allow us to talk about the supports of elements of F ∞ in the language. For that we add constants { S γ : γ ∈ F ∞ } to the signature L ∞ to get a new signature L ′∞ and we consider the theory A ′ θ consistingof: • The axioms of A θ . • For γ ∈ F ∞ , the axioms: – sup a d ( S γ ∧ t γ ( a ) , t γ ( a )) = 0. – µ ( S γ ) = θ (N γ ).This theory expresses that for γ ∈ F ∞ , the constant S γ must be interpreted as supp γ M in themodel M , as it contains the support by the first axiom and has the same measure by the second one.We need a last definition in order to prove that the theories A θ admit amalgamation for θ hyperfinite: Definition 3.26.
Let M | = A θ , we denote by I M and we call the IRS of M the substructure of M generated by the elements supp γ for γ ∈ Γ.Note that this naming is consistent: let M = M α for a pmp action Γ α y ( X, µ ) of IRS θ . Then I M is isomorphic to the measure algebra I θ associated to the action Γ θ y (Sub(Γ) , θ ) and moreover,the map Stab α : X → Sub(Γ) is a lifting of the inclusion I M ⊆ M . MODEL THEORY OF HYPERFINITE ACTIONS Theorem 3.27.
Let θ be an IRS, then the theory A ′ θ admits amalgamation in the signature L ′∞ .Proof. Let M , M | = A ′ θ and let Z be a common substructure of M and M . Then by definitionof the theory A ′ θ , I θ is a substructure of Z and the inclusions Z ֒ → M and Z ֒ → M send I θ on I M and I M respectively. For the sake of simplicity, we identify Z with its images in M and M ,which implies that I θ , I M and I M are all identified.Let X , X and Z be the respective Stone spaces of M , M and Z (see [Fre02, 321J]) and let µ , µ be the respective inner regular Borel probability measures on X and X . We define an inner regularBorel probability measure ν on X × X as in [BY06, Construction 2.3] as the continuous extension ofthe map defined on cylinders by the formula: ν ( a × a ) = Z Z µ ( a | Z ) µ ( a | Z ) dz for all a ∈ M , a ∈ M . The pmp action F ∞ y ( X × X , ν ) then induces a structure N | = A F ∞ that we call the relativeindependent joining of M and M over Z .The following diagram is indeed commutative: NM M Z It remains to prove that N | = A θ . For that note that ¬ supp γ N = _ { a : ∀ b ⊆ a, γb = b } = _ { a × a : ∀ b ⊆ a × a , γb = b } = _ { a × a : ∀ b ⊆ a ∀ b ⊆ a , γb = b and γb = b } = ¬ supp γ M × ¬ supp γ M = ¬ S Z γ × ¬ S Z γ but the definition of ν implies that ν (cid:16) ¬ S Z γ × M (cid:17) = ν (cid:16) ¬ S Z γ × ¬ S Z γ (cid:17) , so that these two elementsof N are equal. Letting i denote the embedding M ֒ → N , we get the equalities ¬ supp γ N = ¬ S Z γ and therefore supp γ N = i (cid:16) S Z γ (cid:17) = i (cid:16) supp γ M (cid:17) . This being true for any γ ∈ F ∞ , it follows that i maps any finite intersection of supports in M to the corresponding intersection of supports in N ,and since i also preserves the measure, we can conclude that N | = A θ . Theorem 3.28.
Let θ be a hyperfinite IRS. Then the theory A ′ θ eliminates quantifiers in the signature L ′∞ .Proof. We use Lemma 3.23.We just saw that A ′ θ admits amalgamation.For model completeness, take M ⊆ N be two models of A ′ θ and let us prove that M (cid:22) N . Let ϕ (¯ x )be an L ′∞ -formula and ¯ p ∈ M n . Then ϕ (¯ x ) is equivalent to a formula of the form ψ (¯ x, S ¯ γ ) where ψ isa L ∞ -formula, and the constants of the form S γ are preserved under the inclusion M ⊆ N . Therefore,it suffices to apply Theorem 3.19 to ψ and to consider the elements S ¯ γ as parameters added to ¯ p toconclude.As a corollary, we get a class of IRSs θ for which the theory A θ admits quantifier elimination. Corollary 3.29.
The theory of free actions of an amenable group admits amalgamation. Namely,if θ is the Dirac measure δ N for a co-amenable normal subgroup N ≤ F ∞ , then A θ has quantifierelimination. MODEL THEORY OF HYPERFINITE ACTIONS Proof.
Simply note that the support of an element γ ∈ F ∞ in a model of A θ is either 0 (if γ ∈ N ) or 1(if γ / ∈ N ). It follows that the theories A θ and A ′ θ completely coincide, hence the result.For M | = A ∞ and A ⊆ M , we write h A i for the closed subalgebra of M (that is, the substructureof M as a model of AMA) generated by A . Theorem 3.30.
Let M | = A θ and A ⊆ M . Then the definable closure of A in M is h F ∞ A ∪ I M i .Proof. On the one hand, A ⊆ dcl M ( A ) and by Lemma 3.15, for γ ∈ F ∞ , supp γ M ∈ dcl M ( A ). Thuswe get the first inclusion.On the other hand, since A ′ θ expands A θ , the definable closure of A in the theory A θ is containedin the definable closure of A in the theory A ′ θ . Let us compute this definable closure D .First, we notice that the function symbols γ are interpreted by automorphisms and thus any atomic L ∞ -formula with parameters in A is equivalent to an atomic L -formula with parameters in F ∞ A . Thisremark then extends to quantifier free formulas.Then, by Theorem 3.28, any L ′∞ -formula with parameters in A is equivalent to a quantifier free L ′∞ -formula with parameters in A and since we only added constants in L ∞ , it is moreover equivalentto a quantifier free L ∞ -formula with parameters in A ∪ I M .Combining the two latter properties and the fact that dcl( A ) = h A i in the theory AMA, we getthat D = h F ∞ ( A ∪ I M ) i . Furthermore, I M is a substructure and so h F ∞ ( A ∪ I M ) i = h F ∞ A ∪ I M i .Hence the conclusion. We recall some definitions from [BYBHU08].
Definition 3.31.
Let κ be a cardinal. A κ -universal domain for a theory T is a κ -saturated andstrongly κ -homogeneous model of T . If U is a κ -universal domain and A ⊆ U , we say that A is small if | A | < κ . Definition 3.32.
Let U be a κ -universal domain for T . A stable independence relation on U is a relation A | ⌣ C B on triples of small subsets of U satisfying the following properties, for all small A, B, C, D ⊆ U , finite ¯ u, ¯ v ⊆ U and small M (cid:22) U :1. Invariance under automorphisms of U .2. Symmetry: A | ⌣ C B ⇐⇒ B | ⌣ C A .3. Transitivity: A | ⌣ C BD ⇐⇒ A | ⌣ C B ∧ A | ⌣ BC D .4. Finite character: A | ⌣ C B if and only if ¯ a | ⌣ C B for every finite ¯ a ⊆ A .5. Existence:
There exists A ′ such that tp( A ′ /C ) = tp( A/C ) and A ′ | ⌣ C B .6. Local character:
There exists B ⊆ B such that | B | ≤ | T | and ¯ u | ⌣ B B .7. Stationarity of types:
If tp( A/ M ) = tp( B/ M ) and A | ⌣ M C and B | ⌣ M C , thentp( A/ M ∪ C ) = tp( B/ M ∪ C ). Proposition 3.33 ([BYBHU08]) . Let κ > | T | and let U be a κ -universal domain. Then the theory T is stable if and only if there exists a stable independence relation on U , and in this case the stableindependence relation is the independence relation given by non-dividing. MODEL THEORY OF HYPERFINITE ACTIONS
Definition 3.34.
From now on, we write hh A ii for dcl U ( A ).Let A, B, C ⊆ U , we say that A and B are independent over C and we write A | ⌣ C B if we have ∀ a ∈ hh A ii , ∀ b ∈ hh B ii , P hh C ii ( a ) P hh C ii ( b ) = P hh C ii ( a ∧ b ).We will need the following propositions: Proposition 3.35 ([Kal02, Proposition 5.6]) . Let
A, B, C ⊆ U | = A F ∞ . Then we have A | ⌣ C B if andonly if ∀ a ∈ hh A ii , P hh BC ii ( a ) = P hh C ii ( a ) . Proposition 3.36 ([BY06, Lemma 2.7]) . Let θ be a hyperfinite IRS on F ∞ .Let U | = A ′ θ and let M , M be small substructures of U . Let Z be a common substructure of M and M . Let M ∧ M be the substructure of U generated by M and M and define N the relativeindependent joining of M and M over Z as in Theorem 3.27.Then M | ⌣ Z M if and only if M ∧ M ≃ N . Theorem 3.37. If θ is a hyperfinite IRS, the relation of independence | ⌣ defined above is a stable inde-pendence relation when restricted to triples of small subsets, relatively to the theory A θ . Consequently,the theory A θ is stable and the relation | ⌣ agrees with non-dividing on triples of small subsets.Proof. Invariance under automorphisms of U : If ρ is an automorphism of U , by uniqueness ofthe orthogonal projection, we know that P hh ρ ( C ) ii = ρ ◦ P hh C ii ◦ ρ − and therefore P hh C ii ( a ) P hh C ii ( b ) = P hh C ii ( a ∧ b ) ⇔ P hh ρ ( C ) ii ( ρa ) P hh ρ ( C ) ii ( ρb ) = P hh ρ ( C ) ii ( ρ ( a ∧ b )) . Symmetry:
The definition is symmetric.3.
Transitivity:
Let
A, B, C, D be small. First if A | ⌣ C B and A | ⌣ BC D then by Proposition 3.35, for a ∈ hh A ii , we have P hh BCD ii ( a ) = P hh BC ii ( a ) = P hh C ii ( a ) so A | ⌣ C BD .Conversely, suppose that A | ⌣ C BD . Then P hh BCD ii ( a ) = P hh C ii ( a ), but that implies that P hh C ii ( a )is a hh C ii -measurable function such that for all hh BCD ii -measurable function f we have R P hh C ii ( a ) f = R a f . We conclude that P hh BCD ii ( a ) = P hh BC ii ( a ) = P hh C ii ( a ), and therefore that A | ⌣ C C and A | ⌣ BC D .4. Finite character:
It follows from the definition and the continuity of P .5. Existence:
Let
A, B, C be small subsets of U . By Löwenheim-Skolem theorem, let A and B besmall structures such that hh AC ii ⊆ A (cid:22) U and hh BC ii ⊆ B (cid:22) U , and let C = hh C ii . Then A and B are both elementary substructures of U containing I U . It follows that A and B | = A ′ θ whenthe constants S γ are interpreted by supp γ U in either of these models, and C is an L ′∞ -commonsubstructure of A and B , so using Theorem 3.27, we see that the relative independent joining D of A and B over C is a small model of A θ .By saturation and homogeneity of U , we can embed D in U while sending B back to B . Takingthe image of A by this embedding gives us a new copy A ′ of A and a new copy A ′ of A . Finally, A ′ ∧ B ≃ D so by Proposition 3.36 we get that A ′ | ⌣ C B , which in turn implies that A ′ | ⌣ C B . EFERENCES
Local character:
Let ¯ u = ( u , . . . , u n ) ⊆ U be finite. Consider the conditional probabilities P hh B ii ( u i ). These are hh B ii -measurable functions with real values and so there is a countablygenerated σ -subalgebra of hh B ii , say hh B ii where B ⊆ B is countable, for which they are allmeasurable. But then we have P hh B ii ( u i ) = P hh B ii ( u i ), so by Proposition 3.35 ¯ u | ⌣ B B .7. Stationarity of types:
We denote by tp L (¯ x/Y ) the type of a tuple ¯ x over a set of parameters Y inthe language L . In other words, this is the type of ¯ x over Y in the underlying atomless measurealgebra of U .Let A, B, C ⊆ U be small and M (cid:22) U be small. Suppose that tp( A/ M ) = tp( B/ M ), A | ⌣ M C and B | ⌣ M C .We begin by proving that tp L ( A/ hh M ∪ C ii ) = tp L ( B/ hh M ∪ C ii ). Indeed, for a ∈ h A i and b ∈h B i , we have P hh M ∪ C ii ( a ) = P M ( a ) and P hh M ∪ C ii ( b ) = P M ( b ), but by Proposition 3.10 types inAMA can be fully described with conditional probabilities and we have tp L ( A/ M ) = tp L ( B/ M )so we get tp L ( A/ hh M ∪ C ii ) = tp L ( B/ hh M ∪ C ii ).Now Theorem 3.28 implies that tp( A/ M ∪ C ) (resp. tp( B/ M ∪ C )) is determined by the L -typetp L ( h F ∞ A ∪ I U i / hh M ∪ C ii ) (resp. tp L ( h F ∞ B ∪ I U i / hh M ∪ C ii )).Thus, let A ′ = F ∞ A ∪ I U and B ′ = F ∞ B ∪ I U .It is clear that tp( A ′ / M ) = tp( B ′ / M ), A ′ | ⌣ M C and B ′ | ⌣ M C and we can apply what we provedjust above to conclude that tp L ( A ′ / hh M ∪ C ii ) = tp L ( B ′ / hh M ∪ C ii ), that istp L ( h F ∞ A ∪ I U i / hh M ∪ C ii ) = tp L ( h F ∞ B ∪ I U i / hh M ∪ C ii ) , hence the conclusion. References [AGV14] Miklós Abért, Yair Glasner, and Bálint Virág. Kesten’s theorem for Invariant RandomSubgroups.
Duke Mathematical Journal , 163(3):465–488, February 2014.[BY06] Itaï Ben Yaacov. Schrödinger’s cat.
Israel Journal of Mathematics , 153:157–191, 2006.[BYBHU08] Itaï Ben Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov.Model theory for metric structures. In Zoe Chatzidakis, Dugald Macpherson, AnandPillay, and Alex Wilkie, editors,
Model Theory with Applications to Algebra and Analysis ,pages 315–427. Cambridge University Press, Cambridge, 2008.[Dye59] H. A. Dye. On groups of measure preserving transformation. I.
Amer. J. Math. , 81:119–159, 1959.[Ele12] Gabor Elek. Finite graphs and amenability.
Journal of Functional Analysis , 263(9):2593–2614, November 2012.[Fre02] David H. Fremlin.
Measure Theory. Vol. 3: Measure Algebras . Number D. H. Fremlin ;Vol. 3 in Measure theory. Fremlin, Colchester, 1. print edition, 2002. OCLC: 248402938.[Fre13] David H. Fremlin.
Measure Theory. Vol. 4 Pt. 1: Topological Measure Spaces . Fremlin,Colchester, 2. ed edition, 2013. OCLC: 935267275.[FRW11] Matthew Foreman, Daniel J. Rudolph, and Benjamin Weiss. The conjugacy problem inergodic theory.
Ann. of Math. (2) , 173(3):1529–1586, 2011.
EFERENCES
J. Eur. Math. Soc. (JEMS) , 6(3):277–292, 2004.[Gla03] Eli Glasner.
Ergodic Theory Via Joinings , volume 101 of
Mathematical Surveys andMonographs . American Mathematical Society, Providence, RI, 2003.[Kal02] Olav Kallenberg.
Foundations of modern probability . Probability and its Applications(New York). Springer-Verlag, New York, second edition, 2002.[Kec10] Alexander Kechris.
Global Aspects of Ergodic Group Actions , volume 160 of
MathematicalSurveys and Monographs . American Mathematical Society, Providence, Rhode Island,January 2010.[KM04] Alexander S. Kechris and Benjamin D. Miller. II. Amenability and Hyperfiniteness. In
Topics in Orbit Equivalence , volume 1852, pages 7–53. Springer Berlin Heidelberg, Berlin,Heidelberg, 2004.[Orn70] Donald Ornstein. Two Bernoulli shifts with infinite entropy are isomorphic.
Advances inMath. , 5:339–348 (1970), 1970.[OW80] Donald S. Ornstein and Benjamin Weiss. Ergodic theory of amenable group actions, 1:The rohlin lemma.
Bull. Amer. Math. Soc , pages 161 – 164, 1980.[Tuc15] Robin D. Tucker-Drob. Weak equivalence and non-classifiability of measure preservingactions.