Uncountable Hyperfiniteness and The Random Ratio Ergodic Theorem
Nachi Avraham-Re'em George Peterzil


UnLearning from Experience to Avoid Spurious Correlations
    Jeff Mitchell

Queen's University Belfast

School of EEECS

United Kingdom 

jmitchell25@qub.ac.uk





Jesús Martínez del Rincón

Queen's University Belfast

School of EEECS

United Kingdom 

j.martinez-del-rincon@qub.ac.uk
Niall McLaughlin

Queen's University Belfast

School of EEECS

United Kingdom 

n.mclaughlin@qub.ac.uk

    September 9, 2024
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§ ABSTRACT 
We show that the orbit equivalence relation of a free action of a locally compact group is hyperfinite (à la Connes–Feldman–Weiss) precisely when it is hypercompact. This implies an uncountable version of the Ornstein–Weiss Theorem and that every locally compact group admitting a hypercompact probability preserving free action is amenable. We also establish an uncountable version of Danilenko's Random Ratio Ergodic Theorem. From this we deduce the Hopf dichotomy for many nonsingular Bernoulli actions.






§ INTRODUCTION


A well-studied property of countable equivalence relations is the property of being hyperfinite, which means that it is the increasing union of countably many finite equivalence relations, where the finiteness is referring to the cardinality of the classes. Of a particular interest are Borel actions of countable groups whose associated orbit equivalence relation (henceforth OER) is hyperfinite. One of the milestones in this theory is the Ornstein–Weiss Theorem <cit.>, by which OERs arising from nonsingular free actions of amenable groups are measure hyperfinite, i.e. hyperfinite up to a zero measure set. The celebrated Connes–Feldman–Weiss Theorem generalized this to amenable equivalence relations (not necessarily OERs) <cit.>.

The usual notion of hyperfiniteness applies to countable equivalence relations only, thus the aforementioned theorems deal with countable groups. In dealing with general locally compact second countable (henceforth lcsc) groups, Connes, Feldman & Weiss used cross sections <cit.>,[The terminology in the field has evolved in an inconsistent way. What is referred to by Connes, Feldman & Weiss as transversal has evolved to what is nowadays usually called cross section, meaning a set which intersects every class countably many times, while transversal nowadays refers to a set intersecting every class exactly once.] which will be introduced below in Section <ref>. Thus, they called an uncountable OER hyperfinite if its restriction to every lacunary cocompact cross section is hyperfinite.[See Remark <ref> below regarding this definition.] In order to avoid confusion, we will call this property sectional-hyperfinite (see Definition <ref> below). Thus, the Connes–Feldman–Weiss Theorem asserts that every OER of an lcsc amenable group is measure-sectional-hyperfinite <cit.>.

In this work we focus on OERs that arise from Borel free actions of lcsc groups. The classes of such an OER are typically uncountable, but using the freeness of the action there is a natural way to define compactness and hypercompactness by pushing forward the topology from the acting group to the orbits. A precise definition will be presented in Section <ref>. With this natural concept defined, we have the following characterization which is purely in the Borel category:



Let G be an lcsc group and X a free Borel G-space. Then E_G^X is sectional-hyperfinite if and only if it is hypercompact.


In <cit.>, Connes, Feldman & Weiss have proved that in the presence of a measure, being sectional-hyperfinite is independent on the the choice of the cross section up to a null set. From Theorem <ref> we can show that this is true already in the Borel level for orbital equivalence relations of free actions:



Let G be an lcsc group and X a free Borel G-space. If E_G^X restricted to one lacunary cocompact cross section is hyperfinite, then its restriction to every lacunary cocompact cross section is hyperfinite.


When adding a measure, from the Connes–Feldman–Weiss Theorem we obtain an uncountable version of the Ornstein–Weiss Theorem:



Every nonsingular free action of an lcsc amenable group is measure-hypercompact.


We continue to study the asymptotic invariance of hypercompact OERs, presented in Section <ref>, and using it to get an uncountable version of a well-known fact in countable groups (see <cit.> with the Ornstein–Weiss Theorem):



An lcsc group G is amenable if and only if one (hence every) of its probability preserving free G-space is measure-hypercompact.


Hyperfiniteness or hypercompactness of an action provides a natural way to take ergodic averages of a function and study their asymptotic behaviour. This was done by Danilenko <cit.> in the countable case, where he established a Random Ratio Ergodic Theorem for nonsingular actions of countable amenable groups using the Ornstein–Weiss Theorem. In the next we exploit Theorem <ref> in order to establish an uncountable version of Danilenko's theorem. We will make use of a natural notion of a Random Følner sequence that will be presented in Section <ref>.



Let G be an lcsc amenable group and (X,μ) a nonsingular probability G-space (not necessarily free). Then there exists a random Følner sequence (𝒮_0,𝒮_1,…) of G such that for every f∈ L^1(X,μ),

    lim_n→∞∫_𝒮_ndμ∘ g/dμ(x)f(g.x)dλ(g)/∫_𝒮_ndμ∘ g/dμ(x)dλ(g)=𝔼(f|Inv_G(X))(x)

for almost every realization (𝒮_0,𝒮_1,…), both μ-a.e. and in L^1(X,μ).


As it was shown by Hochman <cit.>, when fixing one realization of 𝒮_0⊂𝒮_1⊂… it is not always true that the limit in Theorem <ref> holds for all functions in L^1(X,μ) at once. Nevertheless, it was observed by Danilenko that one may restrict the attention to a countable collection of functions in L^1(X,μ), and obtain the following particularly useful corollary:



Let G be an lcsc amenable group and (X,μ) a nonsingular probability G-space. Then for every countable collection ℒ⊂ L^1(X,μ) there exists a Følner sequence S_0⊂ S_1⊂… of G such that for all f∈ℒ,

    lim_n→∞∫_S_ndμ∘ g/dμ(x)f(g.x)dλ(g)/∫_S_ndμ∘ g/dμ(x)dλ(g)=𝔼(f|Inv_G(X))(x)

both μ-a.e. and in L^1(X,μ).


The Random Ratio Ergodic Theorem in countable groups proved itself useful in ergodic theory in recent years, particularly because it is not limited to probability preserving actions but applies also to nonsingular actions, where the pointwise ergodic theorem is generally unavailable as was shown in <cit.>. The classical Hopf method, originally used by Hopf in the probability preserving category to prove the ergodicity of the geodesic flow, was developed in recent years by Kosloff <cit.> and then by Danilenko <cit.> in the nonsingular category. Originally, Kosloff suggested this method for Ratio Ergodic Theorem countable groups (see <cit.>), and Danilenko, observing that the Random Ratio Ergodic Theorem is sufficient, showed that this method applies for all amenable groups.

The works of Kosloff and Danilenko focus on proving ergodicity in nonsingular Bernoulli and Markov shifts in countable groups. In Section <ref> we will demonstrate the power of this method using Theorem <ref> for proving that certain nonsingular Bernoulli shifts of locally compact groups obey the Hopf dichotomy: they are either totally dissipative or ergodic.



§ FUNDAMENTALS



Throughout this work, G stands for a locally compact second countable (lcsc) group, that is, a Polish group whose topology is locally compact. We will fix once and for all a left Haar measure λ on G. A compact filtration of G is a sequence K_0⊂ K_1⊂… of compact subset of G such that G=K_0∪ K_1∪…. Such a compact filtration is called equicompact if it has the property that every compact set C⊂ G is contained in K_n for all sufficiently large n∈ℕ. By a theorem of Struble <cit.>, every lcsc group G admits a compatible proper metric, that is, a metric on G whose topology is the given topology of G and with respect to which closed ball are compact. In particular, an increasing sequence of balls in such a metric whose radii diverge to +∞ forms an equicompact filtration of G.



  §.§.§ Borel G-Spaces


A Borel G-space is a standard Borel space X (whose σ-algebra is fixed but remains implicit) together with a Borel map

    G× X→ X,  (g,x)↦ g.x,

such that e.x=x for every x∈ X, where e∈ G is the identity element, and gh.x=g.(h.x) for every g,h∈ G and x∈ X. A Borel G-space is called free if g.x≠ x for every g∈ G\{e} and x∈ X.



  §.§.§ Cross Sections



The notion of cross section for a Borel G-space is classical and is known for decades, but more recently it went through some useful improvements. See the survey <cit.>. In the following presentation we mostly follow Slutsky's treatment <cit.>.

Let X be a Borel G-space. A cross section (also called complete section or countable section) for X is a Borel set ℭ⊂ X that intersects every orbit in at most countably many points. A cross section ℭ is called U-lacunary, for some symmetric identity neighborhood U⊂ G, if the action map U×ℭ→ X is injective, i.e. U.w∩ U.z=∅ for all distinct w,z∈ℭ. By a theorem of Kechris <cit.>, every Borel G-space admits a lacunary cross section.

A cross section ℭ of X is called K-cocompact, for some compact identity neighborhood K⊂ G, if K.ℭ=X. By a theorem of C. Conley and L. Dufloux, strengthening the aforementioned theorem of Kechris, every Borel G-space admits a cocompact cross section. See <cit.>. The following version is due to Slutsky <cit.>.



Let G be an lcsc group and X a Borel G-space. For every compact symmetric identity neighborhood U⊂ G, there exists a U-lacunary U^2-cocompact cross section for X.




  §.§.§ Voronoi Tessellations



From a cross section of a free Borel G-space one may define in a standard way a Voronoi tessellation of the space. We present this here following Slutsy <cit.>.

Let X be a free Borel G-space and ℭ a lacunary cross section. Fix a compatible proper metric d on G and, as the action is free, consider the map

    d_o:E_G^X→ℝ_≥ 0,   d_o(x,g.x)=d(e,g).

For x∈ X and r≥ 0 denote the set

    ℭ_r(x):={ w∈ℭ∩ G.x:d_o(x,w)≤ r}.

The lacunarity of ℭ shows that ℭ_r(x) is a finite set and, since ℭ is a cross section, for every x∈ X there is some r≥ 0 for which ℭ_r(x) is nonempty. Consequently, the Borel function

    r_o:X→ℝ_≥0,   r_o(x)=min{ d_o(x,w):(x,w)∈(X×ℭ)∩ E_G^X},

is well-defined. Let us consider the finite set

    ℭ(x):=ℭ_r_o(x)(x)={w∈ℭ∩ G.x:d_o(x,w)=r_o(x)}.


The Voronoi tessellation {T_w:w∈ℭ} will be designed to form a partition of X, where we aim to allocate a point x∈ X to a tile T_w if w∈ℭ(x). While for a given x there can be multiple elements in ℭ(x), there are at most finitely many such elements, so fix a Borel linear ordering ≺ on X (say via a Borel isomorphism of X with ℝ) and define the allocation Borel map

    τ_o:X→ℭ by letting τ_o(x) be the ≺-least element of ℭ(x).

Accordingly, we define the Voronoi tessellation {T_w:w∈ℭ} by

    T_w:={x∈ X:τ_o(x)=w},   w∈ℭ.




  §.§.§ OERs, Smoothness and Idealism


Suppose X is a Borel G-space. Then the orbit equivalence relation associated with X is

    E_G^X:={(x,g.x):x∈ X,g∈ G}⊂ X× X.

Since G is lcsc it is known that E_G^X is Borel, i.e. a Borel subset of X× X, and every class in E_G^X, namely every G-orbit, is Borel (see e.g. <cit.>).

The equivalence relation E_G^X is referred to as the orbit equivalence relation associated with X. It will be convenient for us to relax the notion of orbit equivalence relations as follows:



Let X be a Borel G-space. An orbit equivalence relation (OER) on X is a Borel subequivalence relation of E_G^X.

An OER E⊂ E_G^X is said to be positive if there is a identity neighborhood U⊂ G such that for every x∈ X there exists g∈ G with Ug.x⊂[x]_E. In this case we may call it U-positive.


The following notion of smoothness is central in the theory:


An equivalence relation E on a standard Borel space X is called smooth if there is a Borel function s:X→ Y, for any standard Borel space Y, such that for every x,x'∈ X,

    (x,x')∈ E s(x)=s(x').
 


A smooth equivalence relation is always Borel, as it is the inverse image of the diagonal of Y under the Borel function (x,x')↦(s(x),s(x')).

Given a Borel equivalence relation E on a standard Borel space X, consider the space X/E of E-classes of points in X. We recall that a σ-ideal on a given set is a nonempty collection of subsets which is closed under taking subsets and countable unions.


A Borel equivalence relation E on a standard Borel space X is called idealistic if there is a Borel assignment

    I:C↦ I_C,   C∈ X/E,

assigning to each E-class C a σ-ideal I_C on C such that C∉ I_C. Here, I is being Borel in the sense that for every Borel set A⊂ X× X, the set

    A_I:={x∈ X:(A∩ E)_x∈ I_[x]_E},

is Borel in X, where we denote for a general set S⊂ X× X,

    S_x={x'∈ X:(x,x')∈ S}.



The next proposition is analogous to <cit.>:



Every positive OER is idealistic.



For every [x]_E∈ X/E let I_[x]_E be defined by

    S⊂ I_[x]_Eλ(g∈ G:g.x∈ S)=0, for whatever S⊂[x]_E.

By the positivity of E it is clear that [x]_E∉ I_[x]_E and, from monotonicity and σ-additivity of λ, we also see that I_[x]_E is a σ-ideal. In order to see that I is Borel, we note that for a Borel set A⊂ X× X we have by definition

    A_I={ x∈ X:λ(g∈ G:g.x∈(A∩ E)_x)=0}.

Since (x,g)↦1_A∩ E(x,g.x) is a Borel function, from <cit.> it follows that also x↦λ(g∈ G:g.x∈(A∩ E)_x) is a Borel function, hence A_I⊂ X is a Borel set.


The following characterizations are due to Kechris (see <cit.>). For a comprehensive treatment we refer to <cit.>.



Let E be an idealistic Borel equivalence relation (e.g. an OER) on a standard Borel space X. TFAE:

    
  * E is smooth.
    
  * E admits a Borel selector: a Borel function s:X→ X such that
    
    (x,s(x))∈ E and (x,x')∈ E s(x)=s(x') for all x,x'∈ X.

    
  * E admits a Borel transversal: a Borel subset T⊂ X that intersects every class in exactly one point.

When E is smooth with a Borel transversal T, there can be found a Borel selector for E of the form s:X→ T.




§ HYPERCOMPACT OERS



Recall that a countable equivalence relation E on a standard Borel space X is hyperfinite if it admits a finite filtration, namely a sequence E_0⊂ E_1⊂… of equivalence relations with E=E_0∪ E_1∪… such that each E_n is finite, i.e. each E_n-classes is finite. When E is uncountable, we will give a definition for hyperfiniteness following Connes, Feldman & Weiss <cit.>:



Let G be an lcsc group and X a Borel G-space. We say that E_G^X is sectional-hyperfinite if E_G^X∩(ℭ×ℭ) is hyperfinite for every lacunary cocompact cross section ℭ of X.




The definition of Connes, Feldman & Weiss is not restricted to lacunary cocompact cross sections but to all cross sections. However, they have proved that in nonsingular G-spaces being sectional-hyperfinite is independent on the choice of the cross section up to a null set <cit.>. Thus, for nonsingular free G-spaces, sectional-hyperfiniteness as in Definition <ref> and the Connes–Feldman–Weiss notion of hyperfiniteness are the same. We chose to restrict our attention for lacunary cross sections because it turns out to be the best framework in studying hypercompactness, as it is manifested in Theorem <ref>.


We now came to define hypercompactness, which is the natural uncountable analog to the notions of hyperfiniteness. Suppose X is a Borel G-space and E⊂ E_G^X is an OER. For every x∈ X denote

    G_E(x):={g∈ G:(x,g.x)∈ E}.

Since E is Borel this is a Borel set in G. One can routinely verify that

    G_E(g.x)=G_E(x)g^-1 whenever g∈ G_E(x).

Thus, E is U-positive according to Definition <ref> for some identity neighborhood U⊂ G, if for every x∈ X there is g∈ G with Ug⊂ G_E(x).



Let X be a free Borel G-space.

    
  * An OER E⊂ E_G^X is called compact if there is a compact identity neighborhood K⊂ G such that G_E(x)⊂ K for every x∈ X. In this case we may call it K-compact.
    
  * A compact filtration of E_G^X is an increasing sequence of compact OERs E_0⊂ E_1⊂… with E=E_0∪ E_1∪….

    Such a compact filtration is called equicompact if for every x∈ X, the filtration G_E_0(x)⊂ G_E_1(x)⊂… of G is equicompact.[In the sense we defined in Section <ref>.]
    
  * E_G^X is called hypercompact if it admits an equicompact filtration.




We do not know whether hypercompactness can be defined in a meaningful way for general Borel G-spaces, at least not without significant restrictions on the stabilizers. In fact, unlike the notion of hyperfiniteness which is defined plainly for any countable equivalence relation, the notion of hypercompactness is restricted to orbit equivalence relations and relies on the canonical topology of the acting group. Therefore, we do not know whether one can define a meaningful notion of hypercompactness for general equivalence relations.


[Free transitive actions are hypercompact]
It is an elementary fact that the action of every countable group G on itself is hyperfinite. Indeed, we may define for every n∈ℕ a partition T_n of G into finite subsets of G, each of which of size 2^n. We can do this in such a way that T_n+1 is coarser than T_n (i.e. every element of T_n is contained in an element of T_n+1) for every n∈ℕ. Now letting E_n be the equivalence relation whose classes are the elements of T_n for every n∈ℕ, it is easy to see that E_0⊂ E_1⊂… forms a finite filtration of E_G^G=G× G.

Let us show that E_G^G=G× G is hypercompact for every lcsc group G using essentially the same argument. Fix some compatible proper metric d on G. Start by picking a countable set A:={a_1,a_2,…}⊂ G which is 1-discrete (i.e. d(a_i,a_j)≥ 1 for all distinct a_i,a_j∈ A) and 2-dense (i.e. dist(g,A)<2 for every g∈ G). Form a Voronoi tessellation {T_1,T_2,…} so that T_i consists of points which are 1-distant from a_i, for i=1,2,… (just as described in Section <ref>), from which we obtain the equivalence relation E_0 whose classes are {T_1,T_2,…}. Now for every n∈ℕ let E_n be the equivalence relation whose classes are

    { T_1∪… T_2^n,T_2^n+1∪…∪ T_2^n+1,T_2^n+2∪…∪ T_2^n+2+1,…}.

Since A is uniformly discrete, every ball in G is covered by finitely many of the T_i's, hence E_0⊂ E_1⊂… is an equicompact filtration of E_G^G=G× G.




Let X be a free Borel G-space. If E⊂ E_G^X is a positive OER, then every G-orbit contains at most countably many E-classes.



Let G.x be some G-orbit. Letting G.x/E be the set of E-classes in G.x, for every C∈ G.x/E put G_E(C):={ g∈ G:g.x∈ C}. On one hand, those sets are pairwise disjoint and together form a partition of G_E(x). On the other hand, for every C∈ G.x/E, if we fix any g_C∈ G_E(C) then similarly to (<ref>) we note that G_E(C)=G_E(g_C.x)g_C. Assuming that E is U-positive, we deduce that G_E(C) contains a translation of U. Since G is second countable there are at most countably many disjoint translations of U, hence G.x/E is at most countable.




Let X be a free Borel G-space. If E_G^X is hypercompact then it admits an equicompact filtration consisting of positive OERs.



Fix an equicompact filtration F_0⊂ F_1⊂… of E_G^X. For x∈ X put

    φ_0(x)=inf{ m≥ 0:λ(G_F_m(x))>0}.

Since E_G^X is clearly positive, necessarily φ_0(x)<+∞ for every x∈ X. For every m, since F_m is Borel, from <cit.> it follows that x↦λ(G_F_m(x)) is a Borel function, hence so is φ_0. Define then E_0⊂ E by

    (x,x')∈ E_0(x,x')∈ F_φ_1(x).

It is clear that E_0⊂ X× X is a Borel set. Note that if (x,x')∈ F_m for some m∈ℕ and λ(G_F_m(x))>0, then in light of (<ref>) and the quasi-invariance of λ to multiplication from the right also λ(G_F_m(x'))>0, so it easily follows that φ_0(x)=φ_0(x'). This implies that E_o is an equivalence relation. We also note that G_E_0(x)=G_F_φ_0(x)(x), hence E_0 is compact.

Proceeding by induction, for n≥ 1 put

    φ_n(x)=inf{ m≥φ_n-1(x)+1:λ(G_F_m(x))>0},

and define E_n by

    (x,x')∈ E_n(x,x')∈ F_φ_n(x).

It is then routine to verify that E_0⊂ E_1⊂… forms a positive equicompact filtration of E_G^X.


Let us now formulate our first main result, of which Theorem <ref> and Corollary <ref> are particular cases:



For a free Borel G-space X TFAE:

    
  * E_G^X admits a compact filtration.
    
  * E_G^X is sectional-hyperfinite.
    
  * E_G^X is hypercompact.
    
  * E_G^X restricted to one lacunary cocompact cross section is hyperfinite.



The hard part of Theorem <ref> is the implication (3)(4), and its proof relies on the existence of compact OER with sufficient regularity:



Let X be a free Borel G-space and let U⊂ G be an arbitrary relatively compact identity neighborhood. Fix some U-lacunary K-cocompact cross section ℭ for X, for a relatively compact identity neighborhood K⊂ G (which exists, e.g. for K=U^2, by Theorem <ref>), and let {T_w:w∈ℭ} be the corresponding Voronoi tessellation. Define

    E_U⊂ X× X

to be the equivalence relation whose equivalence classes are {T_w:w∈ℭ}.




For all E_U as in Definition <ref> the following properties hold:

    
  * E_U is a smooth OER of E_G^X, and the allocation map τ_o:X→ℭ of the Voronoi tessellation serves as a Borel selector of E_U.
    
  * E_U is a U-positive and K^2-compact.
    
  * If E_0⊂ E_1⊂… is a compact filtration of E_G^X for which E_0⊂ E_U, then it is an equicompact filtration.




First, E_U is an OER of E_G^X since when (x,x')∈ E_U then x,x'∈ T_w for some w∈ℭ, so x,x'∈ G.w. By the construction of the Voronoi tessellation, the allocation map τ_o:X→ℭ is a Borel selector into ℭ, and thus E_U is smooth. By the U-lacunarity of ℭ we see that E_U is U-positive. By the K-cocompactness of ℭ we see that G_E_U(x)⊂ K^2 for every x∈ X, thus E_U is K^2-compact.

Let us now show the third property. Let E_U⊂ E_0⊂ E_1⊂… be a compact filtration. Abbreviate G_n(·)=G_E_n(·) and [·]_n=[·]_E_n. The key property to get equicompactness is the following:


For every x∈ X there exists n∈ℕ such that G_n(x) contains an identity neighborhood in G.


Proof of the claim: Let x∈ X be arbitrary. Since ℭ_r_o(x)+1(x) is a finite set containing ℭ(x), there exists ϵ=ϵ(x)>0 sufficiently small with

    ℭ_r_o(x)+ϵ(x)=ℭ_r_o(x)(x)=ℭ(x).

Look at B_ϵ/2(e), the open ball of radius ϵ/2 around the identity e∈ G with respect to the compatible proper metric defining the cross section ℭ, and we aim to show that B_ϵ/2(e)⊂ G_n(x) for every sufficiently large n∈ℕ. Let g∈ B_ϵ/2(e) be arbitrary. First note that for whatever w∈ℭ(x) we have

    d_o(g.x,w)≤ d_o(x,w)+d_o(x,g.x)<r_o(x)+ϵ/2,

hence by the definition of r_o we have

    r_o(g.x)<r_o(x)+ϵ/2.

Now pick any w∈ℭ(g.x), namely r_o(g.x)=d_o(g.x,w), and we see that

    d_o(x,w)
       ≤ d_o(x,g.x)+d_o(g.x,w)
       <ϵ/2+r_o(g.x)<r_o(x)+ϵ.

This means that w∈ℭ_r_o(x)+ϵ(x)=ℭ(x), so we deduce that ℭ(g.x)⊂ℭ(x). Now since ℭ(x) is finite and E_0⊂ E_1⊂… is a filtration of E_G^X, there can be found n=n(x)∈ℕ sufficiently large such that [ℭ(x)]_n⊂[x]_n, hence

    g.x∈[ℭ(g.x)]_n⊂[ℭ(x)]_n⊂[x]_n.

We found that B_ϵ/2(e)⊂ G_n(x), completing the proof of the claim. ◊

We now deduce the equicompactness of E_U⊂ E_0⊂ E_1⊂…. Let x∈ X be arbitrary and let C⊂ G be some compact set. For every c∈ C, by the claim we may pick n=n(x,c)∈ℕ such that G_n(c^-1.x) contains some identity neighborhood U. Enlarging n if necessary, we may assume that (x,c.x)∈ E_n, that is c∈ G_n(x). Then by (<ref>) we have

    Uc⊂ G_E_U(c.x)c=G_E_U(x) hence c∈Int(G_n(x)),

so we deduce that C⊂⋃_n∈ℕInt(G_n(x)). Since C is compact there exists n_o∈ℕ such that C⊂ G_n(x) for all n>n_o.



 

(1)(2): Fix a compact filtration E_0⊂ E_1⊂… of E_G^X, and let ℭ be a U-lacunary cross section ℭ for E_G^X (we do not use cocompactness here). Put F_n:=E_n∩(ℭ×ℭ) for n∈ℕ. Clearly F_0⊂ F_1⊂… is a filtration of E_G^X∩(ℭ×ℭ), so we have left to show that F_n is a finite equivalence relation for every n∈ℕ. Indeed, since the action is free, for every x∈ X the set G_F_n(x) is U-discrete and contained in G_E_n(x), which is relatively compact. Thus G_F_n(x) is necessarily finite.

(2)(3): Let U⊂ G be some identity neighborhood, pick a U^2-cocompact U-lacunary cross section ℭ for X, and let E_U⊂ E_G^X be as in Definition <ref>. By the assumption that E_G^X is sectional-hyperfinite, we obtain that E_G^X∩(ℭ×ℭ) is a hyperfinite countable equivalence relations. Let then F_0⊂ F_1⊂… be a finite filtration of E_G^X∩(ℭ×ℭ). With the Borel selector τ_o:X→ℭ of E_U, define for n∈ℕ,

    E_n:={(x,y)∈ X× X:(τ_o(x),τ_o(y))∈ F_n}.

Note that E_n⊂ F_n⊂ E_G^X∩(ℭ×ℭ)⊂ E_G^X for every n∈ℕ. Also note that if (x,x')∈ E_G^X then (τ_o(x),τ_o(x'))∈ E_G^X∩(T× T) hence there exists n∈ℕ such that (τ_o(x),τ_o(x'))∈ F_n hence (x,y)∈ E_n. Thus, we found that E_0⊂ E_1⊂… is a filtration of E_G^X. Since (x,x')∈ E_Uτ_o(x)=τ_o(x') it is clear that E_U⊂ E_0. Let us verify that E_n is compact for each n∈ℕ. Indeed, let x∈ X be arbitrary and note that as F_n is finite there is a finite set A_n(x)⊂ G such that for every g∈ G there is some a∈ A_n(x) such that τ_o(g.x)=τ_o(a.x). Hence

    G_E_n(x)
       =⋃_a∈ A_n(x){ g∈ G:τ_o(g.x)=τ_o(a.x)}
       =⋃_a∈ A_n(x){ g∈ G:(g.x,a.x)∈ E}
       =⋃_a∈ A_n(x)G_E_U(a.x)=⋃_a∈ A_n(x)G_E_U(x)a^-1.

Since G_E_U(x) is relatively compact we deduce that so is G_E_n(x). Finally, since E_G^X admits a compact filtration whose first element is E_U, by Lemma <ref> it is an equicompact filtration, thus E_G^X is hypercompact.

(3)(4): This implication is included in the implication (1)(2).

(4)(1): Let ℭ be a K-cocompact U-lacunary cross section for X such that E_G^X∩(ℭ×ℭ) is hyperfinite. Let E_U and τ_o be as in Definition <ref>. Fix a finite filtration F_0⊂ F_1⊂… of E_G^X∩(ℭ×ℭ) and put

    E_n:={(x,x')∈ X× X:(τ_o(x),τ_o(x'))∈ F_n} ,   n∈ℕ.

Then, following the same reasoning in the proof of (2)(3), one can verify that E_0⊂ E_1⊂… is a compact filtration for E_G^X.




§ MEASURE-HYPERCOMPACT OERS


Let us recall the basic setup of the nonsingular ergodic theory. A standard measure space is a measure space (X,μ) such that X is a standard Borel space and μ is a Borel σ-finite measure on X. A Borel set A⊂ X is said to be μ-null if μ(A)=0 and it is said to be μ-conull if μ(X\ A)=0. A Borel property of the points of X will be said to hold modulo μ or μ-a.e. if the set of points satisfying this property is μ-conull.

A nonsingular G-space is a standard measure space (X,μ) such that X is a Borel G-space and μ is quasi-invariant to the action of G on X; that is, the measures μ∘ g^-1 and μ are mutually absolutely continuous for every g∈ G. A particular case of a nonsingular G-space is measure preserving G-space, in which we further have μ∘ g^-1=μ for every g∈ G. When μ is a probability measure, we stress this by using the terminology nonsingular probability G-space or probability preserving G-space.



Let (X,μ) be a nonsingular G-space. We say that E_G^X is μ-sectional-hyperfinite or μ-hypercompact if there exists a μ-conull G-invariant set X_o⊂ X such that

    E_G^X_o:=E_G^X∩(X_o× X_o)

is sectional hyperfinite or hypercompact, respectively.


When the measure μ is clear in the context, we may call these notions by measure-sectional-hyperfinite or measure-hypercompact. As mentioned in Remark <ref>, Connes, Feldman & Weiss proved that being measure-sectional-hyperfinite with respect to one cross section implies the same for all other cross sections.

The requirement in Definition <ref> that X_o would be G-invariant is necessary in order to define hypercompactness, as this notion is define exclusively for OERs, but is unnecessary to define sectional-hyperfiniteness, since hyperfiniteness is defined for general countable equivalence relations. Note however that also in measure-sectional-hyperfiniteness we may always assume that it is the case that X_o is G-invariant; indeed, otherwise we may replace X_o by the G-invariant set G.X_o, which is a Borel set,[It follows from the Arsenin–Kunugui Theorem; <cit.>, <cit.>.] and of course is also μ-conull, and E_G^G.X_o remains sectional-hyperfinite because every cross section for X_o serves also as a cross section for G.X_o.

In light of this discussion, Corollary <ref> is nothing but a reformulation of the celebrated Connes–Feldman–Weiss Theorem using Theorem <ref>:


Let G be an amenable group and (X,μ) be a nonsingular G-space. By the Connes–Feldman–Weiss Theorem there exists a μ-conull set X_o⊂ X such that E_G^X_o is sectional-hyperfinite. By the above discussion we may assume that X_o is G-invariant, thus E_G^X_0 is a sectional hypercompact OER. Then by Theorem <ref> we deduce that E_G^X_o is hypercompact, thus E_G^X is μ-hypercompact.




§ CONDITIONAL EXPECTATION ON COMPACT OERS



Here we introduce a formula for conditional expectation on the σ-algebra of sets that are invariant to a compact OER. It will be useful both in relating hypercompactness to amenability and in establishing the Random Ratio Ergodic Theorem.

Let X be a Borel G-space. For an OER E⊂ E_G^X we denote by

    Inv(E)

the E-invariant σ-algebra whose elements are the Borel sets in X which are unions of E-classes. Thus, a Borel set A⊂ X is in Inv(E) if and only if for every x∈ X, either the entire E-class of x is in A or that the entire E-class of x is outside A. The E_G^X-invariant σ-algebra will be abbreviated by

    Inv_G(X):=Inv(E_G^X).

Thus, a Borel set A⊂ X is in Inv_G(X) if it is a G-invariant set.

From a given filtration E_0⊂ E_1⊂… of E_G^X, we obtain an approximation of Inv_G(X) by the sequence

    Inv(E_0)⊃Inv(E_1)⊃…,

that satisfies

    Inv_G(X)=Inv(E_0)∩Inv(E_1)∩….


Suppose (X,μ) is a nonsingular G-space. It then has an associated Radon–Nikodym cocycle, which is a Borel function

    ∇:G× X→ℝ_>0,  ∇:(g,x)↦∇_g(x),

that satisfies the cocycle identity

    ∇_gh(x)=∇_g(h.x)∇_h(x),   g,h∈ G, x∈ X,

and has the property

    ∇_g(·)=dμ∘ g/dμ(·) in L^1(X,μ) for each g∈ G.

We mention shortly that the fact that there can be found a version of the Radon–Nikodym cocycle that satisfies the cocycle identity pointwise is certainly non-trivial, but is true due to the Mackey Cocycle Theorem (see e.g. <cit.>).

Using nothing but the Fubini Theorem, the Radon–Nikodym property of ∇ can be put generally in the following formula:

    †   ∬_G× X∇_g(x)f_0(g.x)f_1(x)φ(g)dλ⊗μ(g,x)
             =∬_G× Xf_0(x)f_1(g^-1.x)φ(g)dλ⊗μ(g,x),


    for all Borel functions  f_0,f_1:X→[0,∞) and φ:G→[0,∞).


We introduce the main formula we need for conditional expectation on a compact equivalence relation. For finite OERs in the context of countable acting groups, this formula was mentioned in <cit.>  (cf. the generalized Bayes' law in <cit.>).

Let E⊂ E_G^X be a compact OER. With the Radon–Nikodym cocycle (<ref>), define an operator of measurable functions on X by

    S^E:f↦ S_f^E,   S_f^E(x):=∫_G_E(x)∇_g(x)f(g.x)dλ(g).




For a compact positive OER E⊂ E_G^X, the conditional expectation of every f∈ L^1(X,μ) with respect to Inv(E) has the formula

    𝔼(f|Inv(E))(x)=S_f^E(x)/S_1^E(x)=∫_G_E(x)∇_g(x)f(g.x)dλ(g)/∫_G_E(x)∇_g(x)dλ(g),

for μ-a.e. x∈ X (depending on f).


We will start by a simple lemma that demonstrates that compact OERs are dissipative in nature (cf. <cit.>).


Let E⊂ E_G^X be a compact positive OER. For every f∈ L^1(X,μ) we have S_f^E(x)<+∞ for μ-a.e. x∈ X (depending on f).



Pick a compatible proper metric on G and for r>0 let B_r be the ball of radius r around the identity with respect to this metric. Recalling the identity (<ref>), for every f∈ L^1(X,μ) and every r>0 we have

    ∬_B_r× X∇_g(x)f(g.x)dλ⊗μ(g,x)
        =∬_G× Xf(x)1_B_r(g^-1)dλ⊗μ(g,x)
        =λ(B_r^-1)‖ f‖_L^1(X,μ)<+∞.

Since this holds for every r>0, there exists a μ-conull set A_f such that

    ∫_B_r∇_g(x)f(g.x)dλ(g)<+∞ for every r>0 and every x∈ A_f.

Then for every x∈ A_f, since G_E(x) is relatively compact it is contained in some B_r for some sufficiently large r>0, hence S_f^E(x)<+∞.



Note that for every x∈ X, if g∈ G_E(x) then 1_E(g.x,hg.x)=1_E(x,hg.x) for all h∈ G. Let :G→ℝ_>0 be the modular function of G with respect to λ. From the cocycle property of ∇ it follows that whenever g∈ G_E(x),

    I    S_f^E(g.x)
         =∫_G1_E(g.x,hg.x)∇_h(g.x)f(hg.x)dλ(h)
         =∇_g(x)^-1∫_G1_E(x,hg.x)∇_hg(x)f(hg.x)dλ(h)
         =∇_g(x)^-1(g)∫_G1_E(x,h.x)∇_h(x)f(h.x)dλ(h)
       =∇_g(x)^-1(g)S_f^E(x).

In particular, the function X→ℝ_>0, x↦ S_f^E(x)/S_1^E(x), is E-invariant, i.e. Inv(E)-measurable. From all the above we obtain the following identity:

    II   ∫_Xf(x)dμ(x)
         =∫_Xf(x)S_1^E(x)S_1^E(x)^-1dμ(x)
         =∬_G× X∇_g(x)f(x)1_E(x,g.x)S_1^E(x)^-1dλ⊗μ(g,x)
    (<ref>)     =∬_G× Xf(g^-1.x)1_E(g^-1.x,x)S_1^E(g^-1.x)^-1dλ⊗μ(g,x)
    (<ref>)     =∬_G× Xf(g^-1.x)1_E(g^-1.x,x)∇_g^-1(x)(g)S_1^E(x)^-1dλ⊗μ(g,x)
         =∬_G× Xf(g.x)1_E(g.x,x)∇_g(x)S_1^E(x)^-1dλ⊗μ(g,x)
       =∫_XS_f^E(x)S_1^E(x)^-1dμ(x).

Finally, note that for every E-invariant function ψ∈ L^∞(X,μ) we have

    S_f·ψ^E(x)=S_f^E(x)ψ(x) for μ-a.e. x∈ X,

so the identity (<ref>) when applied to f·ψ implies the identity

    ∫_Xf(x)ψ(x)dμ(x)=∫_XS_f^E(x)S_1^E(x)^-1ψ(x)dμ(x).

Then the E-invariance of S_f^E(x)S_1^E(x)^-1 with the last identity readily imply that S_f^E(·)S_1^E(·)^-1=𝔼(f|Inv(E)) in L^1(X,μ).




§ ASYMPTOTIC INVARIANCE OF HYPERCOMPACT OERS



The following asymptotic invariance property in hyperfinite equivalence relations was proved by Danilenko <cit.> and has been used by him for the random ratio ergodic theorem <cit.>. We will show that Danilenko's proof can be adapted to hypercompact OERs as well.

Let G be an lcsc group with a left Haar measure λ. A compact set S⊂ G is said to be [K,ϵ]-invariant, for some compact set K⊂ G and ϵ>0, if

    λ(g∈ S:Kg⊂ S)>(1-ϵ)λ(S).




Let (X,μ) be a probability preserving G-space. Suppose E_G^X is hypercompact with an equicompact filtration E_0⊂ E_1⊂…. Then for every compact set K⊂ G and ϵ>0, it holds that

    lim inf_n→∞μ(x∈ X:G_E_n(x) is [K,ϵ]-invariant)>1-ϵ.



We first formulate a simple probability fact:



Let W_nW be an L^1-convergent sequence of random variables taking values in [0,1]. Then for every 0<ϵ≤ 1/2,

    if 𝔼(W)>1-ϵ^2 then lim inf_n→∞ℙ(W_n>1-ϵ)>1-ϵ.




For every n, since W_n is taking values in [0,1],

    𝔼(W_n)
       =∫_{ W_n>1-ϵ}W_ndℙ+∫_{ W_n≤1-ϵ}W_ndℙ
       ≤ℙ(W_n>1-ϵ)+(1-ϵ)ℙ(W_n≤1-ϵ)
       =1-ϵℙ(W_n≤1-ϵ).

Since the left hand-side converges to 𝔼(W)>1-ϵ^2 as n→+∞, we deduce that ℙ(W_n≤1-ϵ)<ϵ for every sufficiently large n.



For m∈ℕ let X_m(K)={ x∈ X:K⊂ G_m(x)}. From the equicompactness, X_m(K)↗ X as m↗+∞, so pick m_o∈ℕ such that μ(X_m_o(K))>1-ϵ^2. By Lévy's martingale convergence theorem (see e.g. <cit.>),

    W_n:=𝔼(1_X_m_o(K)|Inv(E_n))𝔼(1_X_m_o(K)|Inv_G(X)):=W.

Since 𝔼(W)=μ(X_m_o)>1-ϵ^2, from Lemma <ref> it follows that there exists n_o∈ℕ, say n_o>m_o, such that μ(W_n>1-ϵ)>1-ϵ for every n>n_o.

Abbreviate G_n(·)=G_E_n(·). For every n>n_o, by the formula for conditional expectation as in Proposition <ref> in the probability preserving case, since G_m_o(x)⊂ G_n(x) and G_n(g.x)=G_n(x)g^-1 for g∈ G_n(x), we get

    W_n(x)
       =λ(g∈ G_n(x):K⊂ G_m_o(g.x))/λ(G_n(x))
       ≤λ(g∈ G_n(x):K⊂ G_n(g.x))/λ(G_n(x))
       =λ(g∈ G_n(x):Kg⊂ G_n(x))/λ(G_n(x))

Hence, for all n>n_o,

    μ(x∈ X:G_n(x) is [K,ϵ]-invariant)≥μ(W_n>1-ϵ)>1-ϵ.



We can now prove Theorem <ref>. First we mention that the setting of Theorem <ref> is never void, since every lcsc group admits a probability preserving free G-space (see e.g. <cit.>). Recall that an lcsc group G is amenable if it admits a Følner sequence, namely a compact filtration S_0⊂ S_1⊂… of G such that for every compact set K⊂ G and every ϵ>0, there is n_o∈ℕ such that S_n is [K,ϵ]-invariant for all n>n_o.


One implication is a particular case of Theorem <ref>. For the other implication, suppose that (X,μ) is a probability preserving G-space and that, up to a μ-null set, E_G^X admits an equicompact filtration E_0⊂ E_1⊂…. Fix an equicompact filtration K_0⊂ K_1⊂… of G, and a sequence ϵ_0,ϵ_1,… of positive numbers with ϵ_0+ϵ_1+…<+∞. From Proposition <ref> together with the Borel–Cantelli Lemma, one deduces that for μ-a.e. x∈ X and every m∈ℕ, G_E_n(x) is [K_m,ϵ_m]-invariant for all but finitely many n∈ℕ. It then follows that G_0(x)⊂ G_1(x)⊂… is a Følner sequence of G for μ-a.e. x∈ X, and in particular G is amenable.




§ TWO RANDOM RATIO ERGODIC THEOREMS



The classical Random Ergodic Theorem of Kakutani regards the asymptotic behaviour of ergodic averages of the form

    1/n∑_k=1^nf(z_k.x),   n=1,2,…,

where x is a point in a G-space, (z_1,z_2,…) is a random walk on G, and f is an integrable function. See e.g. <cit.> and the references therein. Performing a random walk on the acting group enables one the use of the classical ergodic averages along the natural Følner sequence of ℕ. Alternatively, one may propose other methods to pick compactly many group elements at random in each step, and study the asymptotic behaviour of the corresponding ergodic averages. This was done by Danilenko for countable amenable groups <cit.> using the Ornstein–Weiss Theorem about hyperfiniteness, and in the following we extend this method to uncountable amenable groups using Theorem <ref> about hypercompactness.



Let (X,μ) be a nonsingular probability G-space and suppose that E_G^X is hypercompact with an equicompact filtration E_0⊂ E_1⊂…. Then for every f∈ L^1(X,μ),

    lim_n→∞∫_G_E_n(x)∇_g(x)f(g.x)dλ(g)/∫_G_E_n(x)∇_g(x)dλ(g)=𝔼(f|Inv_G(X))(x)

both μ-a.e. and in L^1(X,μ).


It is worth noting that in the setting of Theorem <ref>, while it is called an ergodic theorem, we do not say that the filtration has the Følner property.


Since E_0⊂ E_1⊂… is a filtration of E_G^X modulo μ, Inv(E_0)⊃Inv(E_1)⊃…… is an approximation of Inv_G(X) modulo μ. Then by Lévy's martingale convergence theorem (see e.g. <cit.>), for every f∈ L^1(X,μ) it holds that

    lim_n→∞𝔼(f|Inv(E_n))=𝔼(f|Inv_G(X))

both μ-a.e. and in L^1(X,μ). By the formula for conditional expectation as in Proposition <ref> this is the limit stated in the theorem.


The following theorem was proved by Danilenko <cit.> for countable amenable groups, using the Ornstein–Weiss Theorem. For the general case we will follow the same idea as Danilenko, only that we substitute the Ornstein–Weiss Theorem with Theorem <ref> and use the formulas for conditional expectations as in Proposition <ref>. It is worth mentioning that the general idea of the proof is the same, albeit considerably simpler, to the approach used by Bowen & Nevo in establishing pointwise ergodic theorems for probability preserving actions of non-amenable (countable) groups. See e.g. <cit.>.


A random equicompact filtration of G is a sequence (𝒮_0,𝒮_1,…) such that:

    
  * Each 𝒮_n is a function from an abstract probability space (Ω,ℙ) into the relatively compact sets in G.
    
  * Each 𝒮_n is measurable in the sense that {(ω,g):g∈𝒮_n(ω)} is a measurable subset of Ω× S.  
    
  * 𝒮_0⊂𝒮_1⊂… is ℙ-almost surely an equicompact filtration of G.

A random Følner sequence is a random equicompact filtration (𝒮_0,𝒮_1,…) of G that forms ℙ-almost surely a Følner sequence of G.


Our source for Random Følner filtrations is, of course, hypercompact orbit equivalence relations:



Suppose E_G^X is hypercompact with an equicompact filtration  E_0⊂ E_1⊂…. Then the sequence (𝒮_0,𝒮_1,…) that is defined on (X,μ) by 

    𝒮_n(x)=G_E_n(x)={g∈ G:(x,g.x)∈ E}

forms a random equicompact filtration.



Of course, (𝒮_0(x),𝒮_1(x),…) forms an equicompact filtration of G for every x∈ X. In order to see the measurability, note that whenever E⊂ E_G^X is an OER, and in particular a Borel subset of X× X, then the set {(x,g)∈ X× G:g∈ G_E(x)} is nothing but the inverse image of E under the Borel map

    X× G→ X× X,  (x,g)↦(x,g.x).



Having all these notions defined we can prove now Theorem <ref>:


Pick any probability preserving free G-space (Y,ν) (the existence of such a G-space is well-known; see e.g. <cit.>). Since G is amenable, it follows from Theorem <ref> that (Y,ν) is hypercompact, and in fact there can be found an equicompact filtration F_0⊂ F_1⊂… of E_G^Y. Consider the diagonal nonsingular G-space (X× Y,μ⊗ν), namely with the action of G that is given by g.(x,y)=(g.x,g.y). Note that E_G^X× Y is measure-hypercompact, since it has the equicompact filtration E_0⊂ E_1⊂… given by

    E_n={((x,y),(g.x,g.y)):(y,g.y)∈ F_n}.

Indeed, as the action of G on (Y,ν) is free, it is easy to verify that

    G_n(x,y):=G_E_n(x,y)=G_F_n(y),  (x,y)∈ X× Y.

Also, since G acts on (Y,ν) in a probability preserving way, its Radon-Nikodym cocycle can be taken to be the constant function 1, so we have

    ∇_g^μ⊗ν(x,y)=∇_g(x),  (x,y)∈ X× Y,

where ∇_g^μ⊗ν and ∇_g denote the Radon–Nikodym cocycle (<ref>) of the nonsingular G-spaces (X× Y,μ⊗ν) and (X,μ), respectively. Then for every f∈ L^1(X,μ), applying Theorem <ref> to f⊗ 1∈ L^1(X× Y,μ⊗ν), we obtain

    lim_n→∞∫_G_n(y)∇_g(x)f(g.x)dλ(g)/∫_G_n(y)∇_g(x)dλ(g)=𝔼(f⊗1|Inv(E_G^X× Y))=𝔼(f|Inv(E_G^X))

both μ⊗ν-a.e. and in L^1(X× Y,μ⊗ν). Define the random equicompact filtration (𝒮_0,𝒮_1,…) on the probability space (Y,ν) by 𝒮_n(y)=G_E_n(y), n=0,1,…, which is a random equicompact filtration by Lemma <ref>.

Finally, in order to see that (𝒮_0,𝒮_1,…) is a random Følner sequence, note that since F_0⊂ F_1⊂… is an equicompact filtration of G, then E_0⊂ E_1⊂… is an equicompact filtration of E_G^X× Y. Then, as we have shown in the proof of Theorem <ref>, the equicompactness ensures that for ν-a.e. y∈ Y this is a Følner sequence.




§ HOPF DICHOTOMY IN NONSINGULAR BERNOULLI ACTIONS



An important model in nonsingular ergodic theory is the nonsingular Bernoulli G-space. For a countable group G, this is the Borel G-space {0,1}^G, where the action is given by translating the coordinates, and with a product measure of the form ⊗_g∈ G(p_g,1-p_g). A classical theorem of Kakutani provides a criterion on the summability of (p_g)_g∈ G that completely determines when this action becomes nonsingular, in which case ({0,1}^G,⊗_g∈ G(p_g,1-p_g)) is called a nonsingular Bernoulli shift. This model is well-studied in ergodic theory and its related fields in recent years, and many results are known about its conservativity, ergodicity and other ergodic-theoretical properties. See the survey <cit.>.

When it comes to an lcsc group G, the natural model for nonsingular Bernoulli G-spaces is the nonsingular Poisson suspension.[Nonsingular Poisson suspension which has a dissipative base is referred to as nonsingular Bernoulli G-space, following the terminology in <cit.>.] We will introduced the fundamentals of this model below. The analog of the Kakutani dichotomy was established by Takahashi, and the formula for the Radon–Nikodym derivative was presented by Danilenko, Kosloff & Roy <cit.>. See also the survey <cit.>.

Here we will use the Random Ratio Ergodic Theorem <ref> in order to show that for a large class of nonsingular Bernoulli G-spaces the Hopf Dichotomy holds: they are either totally dissipative or ergodic. Our method follows the main line of the Hopf method in proving ergodicity, which was presented in the nonsingular category by Kosloff <cit.> and extended by Danilenko <cit.>.



  §.§.§ Conservativity and Dissipativity


Let (X,μ) be a nonsingular probability G-space. Then (X,μ) is said to be conservative if it has the following recurrence property: for every Borel set A⊂ X with μ(A)>0 and every compact set K⊂ G, there exists g∈ G\ K such that μ(A∩ g.A)>0. It is a classical fact (for a proof of the general case see <cit.>) that conservativity is equivalent to that

    ∫_Gdμ∘ g/dμ(x)dλ(g)=+∞ for μ-a.e. x∈ X.

Accordingly, the Hopf Decomposition or the Conservative–Dissipative Decomposition of (X,μ) is given by

    𝒟:={ x∈ X:∫_Gdμ∘ g/dμ(x)dλ(g)<+∞} and 𝒞:=Y\𝒟.

Thus, (X,μ) is called totally dissipative if μ(𝒟)=1 and, on the other extreme, it is called conservative if μ(𝒞)=1.



  §.§.§ Nonsingular Bernoulli G-spaces


Let G be an lcsc group and μ be an absolutely continuous (i.e. in the Haar measure class) measure on G. Denote the corresponding Radon–Nikodym derivative by

    ∂:G→ℝ_>0,  ∂(x)=dμ/dλ(x).

Thus, (G,μ) is a nonsingular G-space in the action of G on itself by translations, and we have the Radon–Nikodym cocycle

    ∇_g(x):=dμ∘ g/dμ(x)=dμ/dλ(gx)dλ/dμ(x)=∂(gx)/∂(x),   g,x∈ G.

Here μ∘ g is the measure A↦μ(gA) (g forms an invertible transformation).

Let G^∗ be the space of counting Radon measures on G. Thus an element p∈ G^∗ is a Radon measure on G that takes nonnegative integer values. The Borel σ-algebra of G^∗ is generated by the mappings

    N_A:G^∗→ℤ_≥ 0∪{+∞},   N_A(p)=p(A),

for all Borel sets A⊂ G. Denote by μ^∗ the unique probability measure on G^∗ with respect to which the distribution of each of the random variables N_A is Poisson with mean μ(A), meaning that

    μ^∗(N_A=k)=e^-μ(A)μ(A)^k/k!,   k∈ℤ_≥0.

It is a basic fact that when G is non-discrete and μ is absolutely continuous, for every p in a μ^∗-conull set in G^∗ there are no repetitions, that is to say, every such p is of the form p=∑_p∈ Pδ_p, where P⊂ G is some discrete set that we will denote P=Supp(p) and refer to as the support of p.

There is a natural action of G on G^∗, which is given by

    g.p(A)=p(gA),   g∈ G.

By <cit.> (see also <cit.>), when μ satisfies

    ∇_g-1∈ L^1(G,μ) for all g∈ G,

or, equivalently,

    g.∂-∂∈ L^1(G,λ) for all g∈ G,

then the action of G on (G^∗,μ^∗) is nonsingular, and the Radon–Nikodym derivatives have the formula

    ∇_g^∗(p):=dμ^∗∘ g/dμ^∗(p)
       =e^-∫_G(∇_g-1)dμ·∏_x∈Supp(p)∇_g(x) 
       =e^-∫_G(g.∂-∂)dλ·∏_x∈Supp(p)∂(gx)/∂(x),   g∈ G.




  §.§.§ A Hopf Dichotomy


The following theorem is a general result about nonsingular Bernoulli G-spaces. For constructions of nonsingular Bernoulli G-spaces with a variety of ergodic properties we refer to <cit.>.



Let G be an lcsc amenable group and μ an absolutely continuous measure on G such that ∂:=dμ/dλ satisfies

    0<inf∂<sup∂<+∞ and g.∂-∂∈ L^1(G,λ),   g∈ G.

Then the nonsingular G-space (G^∗,μ^∗) is either totally dissipative or ergodic.


The first part of the proof, namely that (G^∗,μ^∗) is either totally dissipative or conservative, is essentially the same as the proof of <cit.>. The main part of the proof, namely that conservativity implies ergodicity, uses Corollary <ref> of the Random Ratio Ergodic Theorem. Both parts relies on the following observation of Danilenko, Kosloff & Roy.

Denote by Π=Π(G,μ) the group of all μ-preserving invertible compactly supported transformations of (G,μ). We denote by Supp(π) the (compact) support of π∈Π. Then Π acts naturally on (G^∗,μ^∗) in a measure preserving way via

    π.p(A)=p(π^-1(A)),  π∈Π, p∈ G^∗.

As it was proved in the course of the proof <cit.>, we have:


The action of Π=Π(G,μ) on (G^∗,μ^∗) is ergodic.



First, note that for every g∈ G we have

    ∫_G|g.∂/∂-1|dμ=∫_G|g.∂-∂|dλ.

Then since g.∂-∂∈ L^1(G,λ) for every g∈ G, it follows from <cit.>) that (G^∗,μ^∗) is a nonsingular G-space.



  §.§.§ Part 1


Look at the dissipative part of (G^∗,μ^∗), that we denote

    𝒟^∗:={ p∈ G^∗:∫_G∇_g^∗(p)dλ(g)<+∞}.

Let us show that 𝒟^∗ is Π-invariant. Find 0<α<1 with α≤inf∂<sup∂≤α^-1. For every π∈Π consider the function

    υ_π:G^∗→ℝ_>0,  υ_π(p)=α^#(Supp(p)∩Supp(π))=α^N_Supp(π)(p),

where P is the support of p and Supp(π) is the support of π. Since Supp(π) is compact, for μ^∗-a.e. p∈ G^∗ we have N_Supp(π)(p)<+∞. By the formula (<ref>) we deduce

    ∇_g^∗(π.p)/∇_g^∗(p) =
    ∏_x∈Supp(p)∩Supp(π)∇_g(π(x))/∇_g(x)∈[υ_π(p)^4, υ_π(p)^-4].

This readily implies the Π-invariance of 𝒟^∗, and since Π acts ergodically we deduce that (G^∗,μ^∗) is either totally dissipative or conservative.



  §.§.§ Part 2


We use Corollary <ref> of the Random Ratio Ergodic Theorem in order to establish the following property: Assuming that (G^∗,μ^∗) is conservative, then for every π∈Π there is a function Υ_π:G^∗→ℝ_>0 such that every nonnegative function φ∈ L^1(G^∗,μ^∗) and μ^∗-a.e. p∈ G^∗,

    Υ_π(p)·𝔼(φ|Inv_G(G^∗))(p)≤𝔼(φ|Inv_G(G^∗))(π.p)≤𝔼(φ|Inv_G(G^∗))(p)/Υ_π(p)

In fact, we will show that Υ_π(p):=υ_π(p)^16=α^16· N_Supp(π)(p) works.

Let φ∈ L^1(G^∗,μ^∗) be some nonnegative function. By a standard approximation argument, in order to establish (<ref>) it is sufficient to assume that φ is uniformly continuous in the vague topology on G^∗, which induces its Borel structure. For such φ, for every π∈Π one can easily verify that for all f∈ C_0(G) (i.e. continuous and vanishes as g→∞),

    ∫_Gf(gπ(x))dp(x)-∫_Gf(gx)dp(x)→0 as g→∞.

That is to say, d(g.π.p,g.p)→0 as g→∞, where d denotes the metric of the vague topology on G^∗. Then by the uniform continuity of φ we obtain

    φ(g.π.p)-φ(g.p)→0 as g→∞ uniformly in p.

Using Corollary <ref>, there exists a Følner sequence S_0⊂ S_1⊂… of G, corresponding to ℒ={φ}, whose ergodic averages

    A_nφ(p):=∫_S_n∇_g^∗(p)φ(g.p)dλ(g)/∫_S_n∇_g^∗(p)dλ(g),   p∈ G^∗, n∈ℕ,

satisfy that

    lim_n→∞A_nφ(p)=𝔼(φ|Inv_G(G^∗))(p) for μ^∗-a.e. p∈ G^∗.

From (<ref>), for μ^∗-a.e. p∈ G^∗ and every n∈ℕ we have

    0.9υ_π(p)^16·∫_S_n∇_g^∗(p)φ(g.π.p)dλ(g)/∫_S_n∇_g^∗(p)dλ(g)≤ A_nφ(π.p)≤υ_π(p)^-16·∫_S_n∇_g^∗(p)φ(g.π.p)dλ(g)/∫_S_n∇_g^∗(p)dλ(g).

Recall that by the conservativity of the action of G on (G^∗,μ^∗) we have that

    lim_n→+∞∫_S_n∇_g^∗(p)dλ(g)=∫_G∇_g^∗(p)dλ(g)=+∞,

so using (<ref>) we deduce that

    lim_n→∞|∫_S_n∇_g^∗(p)φ(g.π.p)dλ(g)/∫_S_n∇_g^∗(p)dλ(g)-A_nφ(p)|
               ≤lim_n→∞∫_S_n∇_g^∗(p)|φ(g.π.p)-φ(g.p)|dλ(g)/∫_S_n∇_g^∗(p)dλ(g)=0.

Finally, when taking the limit as n→+∞ in (<ref>), together with (<ref>) we obtain (<ref>) for the function Υ_π(p)=υ_π(p)^16.



  §.§.§ Part 3


We finally deduce that if (G^∗,μ^∗) is conservative then it is ergodic. If E⊂ G^∗ is a G-invariant set with μ^∗(E)>0, from (<ref>) for the indicator φ=1_E we obtain that 1_E(π.p)=1_E(p) for every π∈Π and μ^∗-a.e. p∈ G^∗, thus E is a Π-invariant. Since Π acts ergodically we deduce μ^∗(E)=1.




 §.§ Acknowledgments


We would like to express our gratitude to Michael Björklund for many fruitful discussions regarding this work, and to Zemer Kosloff for suggesting that an uncountable version of Danilenko's Random Ratio Ergodic Theorem should be true, and for providing us with useful comments and corrections.

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