scholarly journals Equipment of Sets with Cardinality of the Continuum by Structures of Polish Groups with Haar Measures

2016 ◽  
Vol 5 ◽  
pp. 8-22
Author(s):  
Gogi Rauli Pantsulaia
Keyword(s):  
Haar Measure ◽  
Polish Space ◽  
Borel Probability Measure ◽  
Locally Compact ◽  
Polish Groups ◽  
Borel Measures ◽  
Isolated Points ◽  
Borel Probability ◽  
Group Structures ◽  
The Continuum

It is introduced a certain approach for equipment of sets with cardinality of the continuum by structures of Polish groups with two-sided (left or right) invariant Haar measures. By using this approach we answer positively Maleki’s certain question (2012) what are the real k-dimensional manifolds with at least two different Lie group structures that have the same Haar measure. It is demonstrated that for each diffused Borel probability measure defined in a Polish space (G;ρ;Bρ(G)) without isolated points there exist a metric ρ1and a group operation ⊙ in G such that Bρ(G) = Bρ1(G) and (G;ρ1;Bρ1(G);⊙) stands a compact Polish group with a two-sided (left or right) invariant Haar measure μ , where Bρ(G) and Bρ1(G) denote Borel σ-algebras of subsets of G generated by metrics ρ and ρ1, respectively. Similar approach is used for a construction of locally compact non-compact or non-locally compact Polish groups equipped with two-sided (left or right) invariant quasi-finite Borel measures.

scholarly journals Cardinal invariants of Haar null and Haar meager sets

2020 ◽  
pp. 1-27
Author(s):  
Márton Elekes ◽  
Márk Poór
Keyword(s):  
Borel Probability Measure ◽  
Invariant Metric ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Borel Set ◽  
Polish Groups ◽  
Polish Group ◽  
Cardinal Invariants ◽  
Borel Probability ◽  
Meager Sets

A subset X of a Polish group G is Haar null if there exists a Borel probability measure μ and a Borel set B containing X such that μ(gBh) = 0 for every g, h ∈ G. A set X is Haar meager if there exists a compact metric space K, a continuous function f : K → G and a Borel set B containing X such that f−1(gBh) is meager in K for every g, h ∈ G. We calculate (in ZFC) the four cardinal invariants (add, cov, non, cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb {Z}^\omega$ . In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Vidnyánszky.

Homeomorphic Sets of Remote Points

1971 ◽  
Vol 23 (3) ◽  
pp. 495-502 ◽  
Author(s):  
R. Grant Woods
Keyword(s):  
Hausdorff Space ◽  
Continuum Hypothesis ◽  
Locally Compact ◽  
Remote Point ◽  
Completely Regular ◽  
Isolated Points ◽  
Discrete Subspace ◽  
Remote Points ◽  
Separable Metric Space ◽  
The Continuum

Let X be a completely regular Hausdorff space, and let βX denote the Stone-Čech compactification of X. A point p ∈ βX is called a remote point of βX if p does not belong to the βX-closure of any discrete subspace of X. Remote points were first defined and studied by Fine and Gillman, who proved that if the continuum hypothesis is assumed then the set of remote points of βR((βQ) is dense in βR – R(βQ – Q ) (R denotes the space of reals, Q the space of rationals). Assuming the continuum hypothesis, Plank has proved that if X is a locally compact, non-compact, separable metric space without isolated points, then βX has a set of remote points that is dense in βX – X. Robinson has extended this result by dropping the assumption that X is separable.

Actions of non-compact and non-locally compact Polish groups

2000 ◽  
Vol 65 (4) ◽  
pp. 1881-1894 ◽  
Author(s):  
Sławomir Solecki
Keyword(s):  
Equivalence Relation ◽  
Polish Space ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Borel Equivalence Relations ◽  
Polish Groups ◽  
Local Compactness ◽  
Polish Group

AbstractWe show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces.

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 Another proof of Hurewicz theorem

2011 ◽  
Vol 49 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Miroslav Repický
Keyword(s):  
Elementary Proof ◽  
Polish Space ◽  
Closed Sets ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Isolated Points ◽  
The Continuum

ABSTRACT . A Hurewicz theorem says that every coanalytic non-Gδ set C in a Polish space contains a countable set Q without isolated points such that Q̅ ∩ C = Q. We present another elementary proof of this theorem and generalize it for k-Suslin sets. As a consequence, under Martin’s Axiom, we obtain a characterization of ∑12 sets that are the unions of less than the continuum closed sets.

Lp(G)-isotone measures

1975 ◽  
Vol 78 (3) ◽  
pp. 471-481 ◽  
Author(s):  
Beryl J. Peers
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Equivalence Classes ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Borel Measures ◽  
Integrable Functions ◽  
Locally Compact Topological Group

Let G be a locally compact topological group with left Haar measure, m; let M(G) denote the bounded regular Borel measures on G and let Lp(G) denote the equivalence classes of pth power integrable functions on G with respect to the left Haar measure.

A Note on the Vague Topology for Measures

1962 ◽  
Vol 58 (2) ◽  
pp. 421-422 ◽  
Author(s):  
Aubrey Wulfsohn
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
Probability Measures ◽  
Locally Compact ◽  
Borel Measures ◽  
Positive Elements ◽  
Markov Theorem ◽  
Borel Probability ◽  
Locally Compact Hausdorff Space ◽  
Null Set

We refer for general background to N. Bourbaki, Intégration, chapters iv and v. We consider a locally compact Hausdorff space R and denote the set of continuous functions with compact support by The Riesz-Markov theorem shows that there is a 1−1 correspondence between the set of regular Borel measures on R and the set of positive elements of the topological dual of . Let {μn}, μ be regular Borel probability measures on R. The sequence of measures {μn} is said to converge vaguely to μ if, for all . Thus the vague topology is that of simple convergence in . We shall call a μ-measurable set μ-quarrable if its boundary is a μ-null set.

scholarly journals An autocorrelation and a discrete spectrum for dynamical systems on metric spaces

2019 ◽  
pp. 1-17
Author(s):  
DANIEL LENZ
Keyword(s):  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Metric Spaces ◽  
Borel Probability Measure ◽  
Almost Periodic Function ◽  
Almost Periodic ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Aperiodic Order ◽  
Borel Probability

We study dynamical systems $(X,G,m)$ with a compact metric space $X$ , a locally compact, $\unicode[STIX]{x1D70E}$ -compact, abelian group $G$ and an invariant Borel probability measure $m$ on $X$ . We show that such a system has a discrete spectrum if and only if a certain space average over the metric is a Bohr almost periodic function. In this way, this average over the metric plays, for general dynamical systems, a similar role to that of the autocorrelation measure in the study of aperiodic order for special dynamical systems based on point sets.

scholarly journals On Polish groups admitting non-essentially countable actions

2020 ◽  
pp. 1-15
Author(s):  
ALEXANDER S. KECHRIS ◽  
MACIEJ MALICKI ◽  
ARISTOTELIS PANAGIOTOPOULOS ◽  
JOSEPH ZIELINSKI
Keyword(s):  
Equivalence Relation ◽  
Isometry Group ◽  
Alternative Proof ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Polish Groups ◽  
Infinite Dimensional ◽  
Open Question ◽  
New Criterion

Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

A minimal distal map on the torus with sub-exponential measure complexity

2018 ◽  
Vol 40 (4) ◽  
pp. 953-974 ◽  
Author(s):  
WEN HUANG ◽  
LEIYE XU ◽  
XIANGDONG YE
Keyword(s):  
Dynamical System ◽  
Probability Measure ◽  
Borel Probability Measure ◽  
Skew Product ◽  
Topological Dynamical System ◽  
Borel Probability ◽  
Product Map

In this paper the notion of sub-exponential measure complexity for an invariant Borel probability measure of a topological dynamical system is introduced. Then a minimal distal skew product map on the torus with sub-exponential measure complexity is constructed.

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