The restriction of a Borel equivalence relation to a sparse set

2003 ◽  
Vol 42 (4) ◽  
pp. 335-347
Author(s):  
Howard Becker
Keyword(s):  
Equivalence Relation ◽  
Borel Equivalence Relation ◽  
Sparse Set

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.

Smooth Sets for a Borel Equivalence Relation

1995 ◽  
Vol 347 (6) ◽  
pp. 2025
Author(s):  
Carlos E. Uzcategui A.
Keyword(s):  
Equivalence Relation ◽  
Borel Equivalence Relation

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.

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.

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.

Bi-Borel reducibility of essentially countable Borel equivalence relations

2005 ◽  
Vol 70 (3) ◽  
pp. 979-992 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Borel Space ◽  
Borel Equivalence Relations ◽  
Borel Equivalence Relation ◽  
Standard Borel Space ◽  
Borel Reducibility ◽  
New Feature ◽  
Borel Probability ◽  
Image Position

This note answers a questions from [2] by showing that considered up to Borel reducibility, there are more essentially countable Borel equivalence relations than countable Borel equivalence relations. Namely:Theorem 0.1. There is an essentially countable Borel equivalence relation E such that for no countable Borel equivalence relation F (on a standard Borel space) do we haveThe proof of the result is short. It does however require an extensive rear guard campaign to extract from the techniques of [1] the followingMessy Fact 0.2. There are countable Borel equivalence relationssuch that:(i) eachExis defined on a standard Borel probability space (Xx, μx); each Ex is μx-invariant and μx-ergodic;(ii) forx1 ≠ x2 and A μxι -conull, we haveExι/Anot Borel reducible toEx2;(iii) if f: Xx → Xxis a measurable reduction ofExto itself then(iv)is a standard Borel space on which the projection functionis Borel and the equivalence relation Ê given byif and only ifx = x′ andzExz′ is Borel;(V)is Borel.We first prove the theorem granted this messy fact. We then prove the fact.(iv) and (v) are messy and unpleasant to state precisely, but are intended to express the idea that we have an effective parameterization of countable Borel equivalence relations by points in a standard Borel space. Examples along these lines appear already in the Adams-Kechris constructions; the new feature is (iii).Simon Thomas has pointed out to me that in light of theorem 4.4 [5] the Gefter-Golodets examples of section 5 [5] also satisfy the conclusion of 0.2.

scholarly journals Amenable versus hyperfinite Borel equivalence relations

1993 ◽  
Vol 58 (3) ◽  
pp. 894-907 ◽  
Author(s):  
Alexander S. Kechris
Keyword(s):  
Probability Measure ◽  
Equivalence Relation ◽  
Borel Probability Measure ◽  
Amenable Group ◽  
Countable Group ◽  
Equivalence Relations ◽  
Borel Space ◽  
Borel Equivalence Relations ◽  
Borel Equivalence Relation ◽  
Standard Borel Space

LetXbe a standard Borel space (i.e., a Polish space with the associated Borel structure), and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every (Borel) probability measure μ onX, there is a Borel invariant setY⊆Xwith μ(Y) = 1 such thatE↾Y(= the restriction ofEtoY) is hyperfinite. (Recall that a countable group G isamenableif it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG(and more—see below). We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. (As usual, a mapg:Z→W, withZandWstandard Borel spaces, is calleduniversally measurableif it is μ-measurable for every probability measure μ onZ.) Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒ(x)Fƒ(y). We have the following theorem.

Borel equivalence relations which are highly unfree

2008 ◽  
Vol 73 (4) ◽  
pp. 1271-1277 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Relation ◽  
Probability Space ◽  
Ergodic Measure ◽  
Equivalence Relations ◽  
Borel Equivalence Relations ◽  
Borel Equivalence Relation ◽  
Borel Probability

AbstractThere is an ergodic, measure preserving, countable Borel equivalence relation E on a standard Borel probability space (X, μ) such that E∣c is not essentially free on any conull C ⊂ X.

scholarly journals Cofinal families of Borel equivalence relations and quasiorders

2005 ◽  
Vol 70 (4) ◽  
pp. 1325-1340 ◽  
Author(s):  
Christian Rosendal
Keyword(s):  
Equivalence Relation ◽  
Metric Spaces ◽  
Equivalence Relations ◽  
Borel Equivalence Relations ◽  
Compact Metric Spaces ◽  
Borel Equivalence Relation ◽  
Borel Ideal ◽  
Borel Reducibility

AbstractFamilies of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering. ≤B, are constructed. There is an analytic ideal on ω generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spaces, are shown to be Kσ complete.

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.

Sign in / Sign up

Export Citation Format

Share Document