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.

scholarly journals Haar Null Sets and the Consistent Reflection of Non-meagreness

2014 ◽  
Vol 66 (2) ◽  
pp. 303-322 ◽  
Author(s):  
Márton Elekes ◽  
Juris Steprāns
Keyword(s):  
Probability Measure ◽  
Continuum Hypothesis ◽  
Borel Probability Measure ◽  
Baire Category ◽  
Borel Set ◽  
Polish Group ◽  
Borel Probability ◽  
Haar Null Sets ◽  
Definition Of ◽  
Null Set

AbstractA subset X of a Polish group G is called Haar null if there exist a Borel set B ⊃ X and Borel probability measure μ on G such that μ(gBh) = 0 for every g; h ∊ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure μ such that μ (X + t) = 0 for every t ∊ R. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set B. (The answer was already known assuming the Continuum Hypothesis.)This result motivates the following Baire category analogue. It is consistent with ZFC that there exist an abelian Polish group G and a Cantor set C ⊂ G such that for every non-meagre set X ⊂ G there exists a t ∊ G such that C ∩ (X + t) is relatively non-meagre in C. This essentially generalizes results of Bartoszyński and Burke–Miller.

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 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.

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.

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.

Study on Strong Sensitivity of Systems Satisfying the Large Deviations Theorem

2021 ◽  
Vol 31 (10) ◽  
pp. 2150151
Author(s):  
Risong Li ◽  
Tianxiu Lu ◽  
Xiaofang Yang ◽  
Yongxi Jiang
Keyword(s):  
Probability Measure ◽  
Characteristic Function ◽  
Large Deviations ◽  
Open Subset ◽  
Metric Space ◽  
Borel Probability Measure ◽  
Compact Metric Space ◽  
Full Support ◽  
Upper Density ◽  
Borel Probability

Let [Formula: see text] be a nontrivial compact metric space with metric [Formula: see text] and [Formula: see text] be a continuous self-map, [Formula: see text] be the sigma-algebra of Borel subsets of [Formula: see text], and [Formula: see text] be a Borel probability measure on [Formula: see text] with [Formula: see text] for any open subset [Formula: see text] of [Formula: see text]. This paper proves the following results : (1) If the pair [Formula: see text] has the property that for any [Formula: see text], there is [Formula: see text] such that [Formula: see text] for any open subset [Formula: see text] of [Formula: see text] and all [Formula: see text] sufficiently large (where [Formula: see text] is the characteristic function of the set [Formula: see text]), then the following hold : (a) The map [Formula: see text] is topologically ergodic. (b) The upper density [Formula: see text] of [Formula: see text] is positive for any open subset [Formula: see text] of [Formula: see text], where [Formula: see text]. (c) There is a [Formula: see text]-invariant Borel probability measure [Formula: see text] having full support (i.e. [Formula: see text]). (d) Sensitivity of the map [Formula: see text] implies positive lower density sensitivity, hence ergodical sensitivity. (2) If [Formula: see text] for any two nonempty open subsets [Formula: see text], then there exists [Formula: see text] satisfying [Formula: see text] for any nonempty open subset [Formula: see text], where [Formula: see text] there exist [Formula: see text] with [Formula: see text].

scholarly journals Packing topological entropy for amenable group actions

2021 ◽  
pp. 1-35
Author(s):  
DOU DOU ◽  
DONGMEI ZHENG ◽  
XIAOMIN ZHOU
Keyword(s):  
Topological Entropy ◽  
Borel Probability Measure ◽  
Amenable Group ◽  
Borel Subset ◽  
Entropy Inequality ◽  
Probability Measures ◽  
Local Entropy ◽  
Compact Metric Space ◽  
Borel Probability ◽  
Generic Points

Abstract Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$ , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property.

scholarly journals Bruckner–Garg-Type Results with Respect to Haar Null Sets inC[0, 1]

2016 ◽  
Vol 60 (1) ◽  
pp. 17-30 ◽  
Author(s):  
Richárd Balka ◽  
Udayan B. Darji ◽  
Márton Elekes
Keyword(s):  
Level Sets ◽  
Borel Probability Measure ◽  
Baire Category ◽  
Point Of View ◽  
Occupation Measure ◽  
Borel Set ◽  
Perfect Set ◽  
Perfect Form ◽  
Borel Probability ◽  
Haar Null Sets

AbstractA setisshyorHaar null(in the sense of Christensen) if there exists a Borel setand a Borel probability measureμonC[0, 1] such thatandfor allf∈C[0, 1]. The complement of a shy set is called aprevalentset. We say that a set isHaar ambivalentif it is neither shy nor prevalent.The main goal of the paper is to answer the following question: what can we say about the topological properties of the level sets of the prevalent/non-shy manyf∈C[0, 1]?The classical Bruckner–Garg theorem characterizes the level sets of the generic (in the sense of Baire category)f∈C[0, 1] from the topological point of view. We prove that the functionsf∈C[0, 1] for which the same characterization holds form a Haar ambivalent set.In an earlier paper, Balkaet al. proved that the functionsf∈C[0, 1] for which positively many level sets with respect to the Lebesgue measure λ are singletons form a non-shy set inC[0, 1]. The above result yields that this set is actually Haar ambivalent. Now we prove that the functionsf∈C[0, 1] for which positively many level sets with respect to the occupation measure λ ◦f–1are not perfect form a Haar ambivalent set inC[0, 1].We show that for the prevalentf∈C[0, 1] for the genericy∈f([0, 1]) the level setf–1(y) is perfect. Finally, we answer a question of Darji and White by showing that the set of functionsf∈C[0, 1] for which there exists a perfect setPf⊂ [0, 1] such thatfʹ(x) = ∞ for allx∈Pfis Haar ambivalent.

REMARKS ON THE IMPRIMITIVITY THEOREM FOR NONLOCALLY COMPACT POLISH GROUPS

2000 ◽  
Vol 03 (02) ◽  
pp. 247-262
Author(s):  
FRANCESCO FIDALEO
Keyword(s):  
Invariant Measure ◽  
Probability Measure ◽  
Invariant Probability Measure ◽  
Equivalence Classes ◽  
Fréchet Spaces ◽  
Locally Compact ◽  
Polish Groups ◽  
Polish Group ◽  
One To One

In this paper we analyze the possibility of establishing a Theorem of Imprimitivity in the case of nonlocally compact Polish groups. We prove that systems of imprimitivity for a Polish group G based on a locally compact homogeneous G-space M ≡ G/H equipped with a quasi-invariant probability measure μ, are in one-to-one correspondence with elements of the space [Formula: see text] of the first cohomology of the group G of equivalence classes of continuous cocycles. As a corollary, we have the complete Imprimitivity Theorem [Formula: see text] in the case of discrete countable homogeneous G-spaces equipped with a quasi-invariant measure. Finally, we outline the possibility of establishing the complete Imprimitivity Theorem for particular classes of Polish groups. These examples cover the case of (separable) Fréchet spaces, for which it is shown that the complete Imprimitivity Theorem holds as well.

scholarly journals Central Limit Theorem and the Distribution of Sequences

2020 ◽  
Vol 77 (1) ◽  
pp. 43-52
Author(s):  
Milan Paštéka
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Probability Measure ◽  
Central Limit ◽  
Borel Probability Measure ◽  
Continuous Distribution ◽  
Distribution Functions ◽  
Compact Metric Space ◽  
The Central Limit Theorem ◽  
Borel Probability

AbstractThe paper deals with independent sequences with continuous asymptotic distribution functions. We construct a compact metric space with Borel probability measure. We use its properties to prove the central limit theorem for independent sequences with continuous distribution functions.

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