The existence of measures of a given cocycle, I: atomless, ergodic σ-finite measures

2008 ◽  
Vol 28 (5) ◽  
pp. 1599-1613 ◽  
Author(s):  
BENJAMIN MILLER
Keyword(s):  
Equivalence Relation ◽  
Polish Space ◽  
Finite Measure ◽  
Borel Equivalence Relation ◽  
Almost Everywhere ◽  
Finite Measures

AbstractGiven a Polish space X, a countable Borel equivalence relation E on X, and a Borel cocycle $\rho : E \rightarrow (0, \infty )$, we characterize the circumstances under which there is a suitably non-trivial σ-finite measure μ on X such that, for every Borel injection ϕ whose graph is contained in E, ρ(ϕ−1(x),x)=[d(ϕ*μ)/dμ](x) μ-almost everywhere.

The existence of measures of a given cocycle, II: probability measures

2008 ◽  
Vol 28 (5) ◽  
pp. 1615-1633 ◽  
Author(s):  
BENJAMIN MILLER
Keyword(s):  
Probability Measure ◽  
Equivalence Relation ◽  
Polish Space ◽  
Probability Measures ◽  
Borel Equivalence Relation ◽  
Almost Everywhere

AbstractGiven a Polish space X, a countable Borel equivalence relation E on X, and a Borel cocycle $\rho : E \rightarrow (0, \infty )$, we characterize the circumstances under which there is a probability measure μ on X such that ρ(ϕ−1(x),x)=[d(ϕ*μ)/dμ](x) μ-almost everywhere, for every Borel injection ϕ whose graph is contained in E.

scholarly journals Incomparable actions of free groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2084-2098
Author(s):  
CLINTON T. CONLEY ◽  
BENJAMIN D. MILLER
Keyword(s):  
Probability Measure ◽  
Free Group ◽  
Equivalence Relation ◽  
Polish Space ◽  
Borel Probability Measure ◽  
Free Groups ◽  
Borel Equivalence Relation ◽  
Separability Condition ◽  
Borel Probability ◽  
Free Actions

Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $\unicode[STIX]{x1D707}$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group $\unicode[STIX]{x1D6E4}$, there is a Borel sequence $(\cdot _{r})_{r\in \mathbb{R}}$ of free actions of $\unicode[STIX]{x1D6E4}$ on $X$, generating subequivalence relations $E_{r}$ of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic, with the further property that $(E_{r})_{r\in \mathbb{R}}$ is an increasing sequence of relations which are pairwise incomparable under $\unicode[STIX]{x1D707}$-reducibility. In particular, we show that if $E$ satisfies a natural separability condition, then this is the case as long as there exists a free Borel action of a countable non-abelian free group on $X$, generating a subequivalence relation of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic.

scholarly journals Trees and amenable equivalence relations

1990 ◽  
Vol 10 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Scot Adams
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Measure Space ◽  
Polynomial Growth ◽  
Finite Measure ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Borel Equivalence Relation

AbstractLet R be a Borel equivalence relation with countable equivalence classes on a measure space M. Intuitively, a ‘treeing’ of R is a measurably-varying way of makin each equivalence class into the vertices of a tree. We make this definition rigorous. We prove that if each equivalence class becomes a tree with polynomial growth, then the equivalence relation is amenable. We prove that if the equivalence relation is finite measure-preserving and amenable, then almost every tree (i.e., equivalence class) must have one or two ends.

scholarly journals Countable sections for locally compact group actions

1992 ◽  
Vol 12 (2) ◽  
pp. 283-295 ◽  
Author(s):  
Alexander S. Kechris
Keyword(s):  
Compact Group ◽  
Equivalence Relation ◽  
Group Actions ◽  
Locally Compact Group ◽  
Borel Space ◽  
Locally Compact ◽  
Borel Equivalence Relation ◽  
Measurable Section ◽  
Singular Action

AbstractIt has been shown by J. Feldman, P. Hahn and C. C. Moore that every non-singular action of a second countable locally compact group has a countable (in fact so-called lacunary) complete measurable section. This is extended here to the purely Borel theoretic category, consisting of a Borel action of such a group on an analytic Borel space (without any measure). Characterizations of when an arbitrary Borel equivalence relation admits a countable complete Borel section are also established.

A Metamathematical Condition Equivalent to the Existence of a Complete Left Invariant Metric for a Polish Group

2006 ◽  
Vol 71 (4) ◽  
pp. 1108-1124 ◽  
Author(s):  
Alex Thompson
Keyword(s):  
Equivalence Relation ◽  
Polish Space ◽  
Invariant Metrics ◽  
Orbit Equivalence ◽  
Invariant Metric ◽  
Continuous Action ◽  
Polish Groups ◽  
Polish Group ◽  
Left Invariant Metrics

AbstractStrengthening a theorem of Hjorth this paper gives a new characterization of which Polish groups admit compatible complete left invariant metrics. As a corollary it is proved that any Polish group without a complete left invariant metric has a continuous action on a Polish space whose associated orbit equivalence relation is not essentially countable.

scholarly journals Borel structures and Borel theories

2011 ◽  
Vol 76 (2) ◽  
pp. 461-476 ◽  
Author(s):  
Greg Hjorth ◽  
André Nies
Keyword(s):  
Equivalence Relation ◽  
Equivalence Classes ◽  
Strong Sense ◽  
Borel Equivalence Relation ◽  
And Function

AbstractWe show that there is a complete, consistent Borel theory which has no “Borel model” in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are equivalence classes under some Borel equivalence relation and the interpretations of the relations and function symbols are uniformly Borel.We also investigate Borel isomorphisms between Borel structures.

An almost everywhere central limit theorem

1988 ◽  
Vol 104 (3) ◽  
pp. 561-574 ◽  
Author(s):  
Gunnar A. Brosamler
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Finite Measure ◽  
Standard Normal Distribution ◽  
Almost Everywhere ◽  
Real Functions ◽  
Standard Normal ◽  
Image Position

The purpose of this paper is the proof of an almost everywhere version of the classical central limit theorem (CLT). As is well known, the latter states that for IID random variables Y1, Y2, … on a probability space (Ω, , P) with we have weak convergence of the distributions of to the standard normal distribution on ℝ. We recall that weak convergence of finite measures μn on a metric space S to a finite measure μ on S is defined to mean thatfor all bounded, continuous real functions on S. Equivalently, one may require the validity of (1·1) only for bounded, uniformly continuous real functions, or even for all bounded measurable real functions which are μ-a.e. continuous.

scholarly journals On the almost everywhere convergence of the ergodic averages

1990 ◽  
Vol 10 (1) ◽  
pp. 141-149
Author(s):  
F. J. Martín-Reyes ◽  
A. De La Torre
Keyword(s):  
Measure Space ◽  
Uniform Boundedness ◽  
Finite Measure ◽  
Necessary And Sufficient Condition ◽  
Ergodic Averages ◽  
Measurable Transformation ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Finite Invariant Measure ◽  
Necessary And Sufficient

AbstractLet (X, ν) be a finite measure space and let T: X → X be a measurable transformation. In this paper we prove that the averages converge a.e. for every f in Lp(dν), 1 < p < ∞, if and only if there exists a measure γ equivalent to ν such that the averages apply uniformly Lp(dν) into weak-Lp(dγ). As a corollary, we get that uniform boundedness of the averages in Lp(dν) implies a.e. convergence of the averages (a result recently obtained by Assani). In order to do this, we first study measures v equivalent to a finite invariant measure μ, and we prove that supn≥0An(dν/dμ)−1/(p−1) a.e. is a necessary and sufficient condition for the averages to converge a.e. for every f in Lp(dν).

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