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Jabłońska Eliza - One of the best experts on this subject based on the ideXlab platform.
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Haar-$\mathcal I$ Sets: looking at small Sets in Polish groups through compact glasses
2019Co-Authors: Banakh Taras, Jabłońska Eliza, Głąb Szymon, Swaczyna JarosławAbstract:Generalizing Christensen's notion of a Haar-null Set and Darji's notion of a Haar-Meager Set, we introduce and study the notion of a Haar-$\mathcal I$ Set in a Polish group. Here $\mathcal I$ is an ideal of subSets of some compact metrizable space $K$. A Borel subSet $B\subSet X$ of a Polish group $X$ is called Haar-$\mathcal I$ if there exists a continuous map $f:K\to X$ such that $f^{-1}(B+x)\in\mathcal I$ for all $x\in X$. Moreover, $B$ is generically Haar-$\mathcal I$ if the Set of witness functions $\{f\in C(K,X):\forall x\in X\;\;f^{-1}(B+x)\in\mathcal I\}$ is coMeager in the function space $C(K,X)$. We study (generically) Haar-$\mathcal I$ Sets in Polish groups for many concrete and abstract ideals $\mathcal I$, and construct the corresponding distinguishing examples. We prove some results on Borel hull of Haar-$\mathcal I$ Sets, generalizing results of Solecki, Elekes, Vidny\'anszky, Dole\v{z}al, Vlas\v{a}k on Borel hulls of Haar-null and Haar-Meager Sets. Also we establish various Steinhaus properties of the families of (generically) Haar-$\mathcal I$ Sets in Polish groups for various ideals $\mathcal I$.Comment: 71 page
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Remarks on Haar Meager Sets and Haar null Sets in spaces of sequences
2014Co-Authors: Jabłońska ElizaAbstract:In the paper we will show how to construct a Haar Meager Set (consequently Meager) which is not Haar null, and conversely, a Meager Haar null Set which is not Haar Meager in spaces of sequences: $l_p$ with $p\geq1$, $c_0$ or $c$. It refers to the paper \cite{Darji}
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Some analogies between Haar Meager Sets and Haar null Sets in abelian Polish groups
2014Co-Authors: Jabłońska ElizaAbstract:In the paper we would like to pay attention to some analogies between Haar Meager Sets and Haar null Sets. Among others, we will show that $0\in \inn (A-A)$ for each Borel Set $A$, which is not Haar Meager in an abelian Polish group. Moreover, we will give an example of a Borel non-Haar Meager Set $A\subSet c_0$ such that $\inn (A+A)=\emptySet$. Finally, we will define $D$-measurability as a topological analog of Christensen measurability, and apply our generalization of Piccard's theorem to prove that each $D$-measurable homomorphism is continuous. Our results refer to the papers \cite{Ch}, \cite{Darji} and \cite{FS}
Vlasák Václav - One of the best experts on this subject based on the ideXlab platform.
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Haar Meager Sets, their hulls, and relationship to compact Sets
2016Co-Authors: Doležal Martin, Vlasák VáclavAbstract:Let $G$ be an abelian Polish group. We show that there is a strongly Haar Meager Set in $G$ without any $F_{\sigma}$ Haar Meager hull (and that this still remains true if we replace $F_{\sigma}$ by any other class of the Borel hierarchy). We also prove that there is a coanalytic naively strongly Haar Meager Set without any Haar Meager hull. Further, we investigate the relationship of the collection of all compact Sets to the collection of all Haar Meager Sets in non-locally compact Polish groups.Comment: 18 page
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Haar Meager Sets revisited
2015Co-Authors: Doležal Martin, Rmoutil Martin, Vejnar Benjamin, Vlasák VáclavAbstract:In the present article we investigate Darji's notion of Haar Meager Sets from several directions. We consider alternative definitions and show that some of them are equivalent to the original one, while others fail to produce interesting notions. We define Haar Meager Sets in nonabelian Polish groups and show that many results, including the facts that Haar Meager Sets are Meager and form a $\sigma$-ideal, are valid in the more general Setting as well. The article provides various examples distinguishing Haar Meager Sets from Haar null Sets, including decomposition theorems for some subclasses of Polish groups. As a corollary we obtain, for example, that $\mathbb Z^\omega$, $\mathbb R^\omega$ or any Banach space can be decomposed into a Haar Meager Set and a Haar null Set. We also establish the stability of non-Haar Meagerness under Cartesian product.Comment: 19 page
Anush Tserunyan - One of the best experts on this subject based on the ideXlab platform.
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finite generators for countable group actions finite index pairs of equivalence relations complexity measures for recursive programs
The Bulletin of Symbolic Logic, 2018Co-Authors: Anush TserunyanAbstract:Author(s): Tserunyan, Anush | Advisor(s): Kechris, Alexander S; Neeman, Itay | Abstract: Part 1: Consider a continuous action of a countable group G on a Polish space X. A countable Borel partition P of X is called a generator if the σ-algebra generated by the Set of the G-translates of P is the Borel σ-algebra of X. It was asked by Benjamin Weiss in ’87 whether the nonexistence of an invariant probability measure implies the existence of a finite generator. The main result of this part is obtaining a positive answer to this question in case X is σ-compact (in particular, when X is locally compact). We also show that finite generators always exist modulo a Meager Set, answering positively a question raised by Alexander Kechris in the mid-’90s.Part 2: We investigate pairs of countable Borel equivalence relations (E, F), where E is a finite index subequivalence relation of F. Our main focus is the well-known problem of whether the treeability of E implies that of F: we provide various reformulations of it and reduce it to one natural universal example. In the measure-theoretic context, assuming that F is ergodic, we characterize the case when E is normal. Finally, in the ergodic case, we characterize the equivalence relations that arise from almost free actions of virtually free groups.Part 3: We consider natural complexity measures for recursive programs from given primitives and derive inequalities between them, answering a question asked by Yiannis Moschovakis.
Tserunyan Anush - One of the best experts on this subject based on the ideXlab platform.
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Finite generators for countable group actions; Finite index pairs of equivalence relations; Complexity measures for recursive programs
eScholarship University of California, 2013Co-Authors: Tserunyan AnushAbstract:Part 1: Consider a continuous action of a countable group G on a Polish space X. A countable Borel partition P of X is called a generator if the σ-algebra generated by the Set of the G-translates of P is the Borel σ-algebra of X. It was asked by Benjamin Weiss in ’87 whether the nonexistence of an invariant probability measure implies the existence of a finite generator. The main result of this part is obtaining a positive answer to this question in case X is σ-compact (in particular, when X is locally compact). We also show that finite generators always exist modulo a Meager Set, answering positively a question raised by Alexander Kechris in the mid-’90s.Part 2: We investigate pairs of countable Borel equivalence relations (E, F), where E is a finite index subequivalence relation of F. Our main focus is the well-known problem of whether the treeability of E implies that of F: we provide various reformulations of it and reduce it to one natural universal example. In the measure-theoretic context, assuming that F is ergodic, we characterize the case when E is normal. Finally, in the ergodic case, we characterize the equivalence relations that arise from almost free actions of virtually free groups.Part 3: We consider natural complexity measures for recursive programs from given primitives and derive inequalities between them, answering a question asked by Yiannis Moschovakis
Doležal Martin - One of the best experts on this subject based on the ideXlab platform.
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Haar Meager Sets, their hulls, and relationship to compact Sets
2016Co-Authors: Doležal Martin, Vlasák VáclavAbstract:Let $G$ be an abelian Polish group. We show that there is a strongly Haar Meager Set in $G$ without any $F_{\sigma}$ Haar Meager hull (and that this still remains true if we replace $F_{\sigma}$ by any other class of the Borel hierarchy). We also prove that there is a coanalytic naively strongly Haar Meager Set without any Haar Meager hull. Further, we investigate the relationship of the collection of all compact Sets to the collection of all Haar Meager Sets in non-locally compact Polish groups.Comment: 18 page
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Haar Meager Sets revisited
2015Co-Authors: Doležal Martin, Rmoutil Martin, Vejnar Benjamin, Vlasák VáclavAbstract:In the present article we investigate Darji's notion of Haar Meager Sets from several directions. We consider alternative definitions and show that some of them are equivalent to the original one, while others fail to produce interesting notions. We define Haar Meager Sets in nonabelian Polish groups and show that many results, including the facts that Haar Meager Sets are Meager and form a $\sigma$-ideal, are valid in the more general Setting as well. The article provides various examples distinguishing Haar Meager Sets from Haar null Sets, including decomposition theorems for some subclasses of Polish groups. As a corollary we obtain, for example, that $\mathbb Z^\omega$, $\mathbb R^\omega$ or any Banach space can be decomposed into a Haar Meager Set and a Haar null Set. We also establish the stability of non-Haar Meagerness under Cartesian product.Comment: 19 page