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Free paratopological groups Open
Let FP(X) be the free paratopological group on a topological space X in the sense of Markov. In this paper, we study the group FP(X) on a $P_\alpha$-space $X$ where $\alpha$ is an infinite cardinal and then we prove that the group FP(X) is…
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Decomposing Borel functions using the Shore–Slaman join theorem Open
Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_\sigma $ set under tha…
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GENERIC ORBITS AND TYPE ISOLATION IN THE GURARIJ SPACE Open
International audience
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The tail process and tail measure of continuous time regularly varying\n stochastic processes Open
The goal of this paper is to investigate the tools of extreme value theory\noriginally introduced for discrete time stationary stochastic processes (time\nseries), namely the tail process and the tail measure, in the framework of\ncontinuo…
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Existence of mark functions in marked metric measure spaces Open
We give criteria on the existence of a so-called mark function in the context\nof marked metric measure spaces (mmm-spaces). If an mmm-space admits a mark\nfunction, we call it functionally-marked metric measure space (fmm-space). This\nis…
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POLISH MODELS AND SOFIC ENTROPY Open
We deduce properties of the Koopman representation of a positive entropy probability measure-preserving action of a countable, discrete, sofic group. Our main result may be regarded as a ‘representation-theoretic’ version of Sinaǐ’s factor…
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Invariant idempotent measures Open
The idempotent mathematics is a part of mathematics in which arithmetic operations in the reals are replaced by idempotent operations. In the idempotent mathematics, the notion of idempotent measure (Maslov measure) is a counterpart of the…
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Descriptive Complexity on Non-Polish Spaces Open
Represented spaces are the spaces on which computations can be performed. We investigate the descriptive complexity of sets in represented spaces. We prove that the standard representation of a countably-based space preserves the effective…
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Sequential separability vs selective sequential separability Open
A space X is sequentially separable if there is a countable D ? X such that every point of X is the limit of a sequence of points from D. We present two examples of a sequentially separable space which is not selectively sequentially separ…
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Measure reducibility of countable Borel equivalence relations Open
We show that every basis for the countable Borel equivalence relations\nstrictly above $\\mathbb{E}_0$ under measure reducibility is uncountable,\nthereby ruling out natural generalizations of the Glimm-Effros dichotomy. We\nalso push many…
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DEGREE SPECTRA OF HOMEOMORPHISM TYPE OF COMPACT POLISH SPACES Open
A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a $\mathbf {0}…
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Polish groupoids and functorial complexity Open
We introduce and study the notion of functorial Borel complexity for Polish groupoids. Such a notion aims to measure the complexity of classifying the objects of a category in a constructive and functorial way. In the particular case of pr…
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Games and hereditary Baireness in hyperspaces and spaces of probability\n measures Open
We establish that the existence of a winning strategy in certain topological\ngames, closely related to a strong game of Choquet, played in a topological\nspace $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$\nequipped…
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Tail measures and regular variation Open
A general framework for the study of regular variation (RV) is that of Polish star-shaped metric spaces, while recent developments in [1] have discussed RV with respect to some properly localised boundedness $\mathcal{B}$ imposing weak ass…
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von Neumann’s problem and extensions of non-amenable equivalence relations Open
The goals of this paper are twofold. First, we generalize the result of Gaboriau and Lyons [17] to the setting of von Neumann's problem for equivalence relations, proving that for any non-amenable ergodic probability measure preserving (pm…
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Polish groups and Baire category methods Open
This article is a slightly modified version of the author’s habilitation thesis, presenting his work on topics related to Polish groups, Baire category methods and metric model theory. Nearly all results presented are not new, though some …
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On Polish groups admitting non-essentially countable actions Open
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 po…
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Absorbing Markov decision processes Open
In this paper, we study discrete-time absorbing Markov Decision Processes (MDP) with measurable state space and Borel action space with a given initial distribution. For such models, solutions to the characteristic equation that are not oc…
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On Borel reducibility in generalized Baire space Open
We study the Borel reducibility of Borel equivalence relations on the generalized Baire space $\kappa ^\kappa $ for an uncountable $\kappa $ with $\kappa ^{<\kappa }=\kappa $. The theory looks quite different from its classical counterpart…
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Functional envelopes relative to the point-open topology on a subset Open
If $(X, f)$ is a dynamical system given by a locally compact separable metric space $X$ without isolated points and a continuous map $f : X\to X $ , and $A$ is a countable dense subset of $X$ , then by the functional envelope of $(X, f)$ r…
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Measurable versions of the Lov\'{a}sz Local Lemma and measurable graph colorings. Open
In this paper we investigate the extent to which the Lov\'asz Local Lemma (an important tool in probabilistic combinatorics) can be adapted for the measurable setting. In most applications, the Lov\'asz Local Lemma is used to produce a fun…
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Extreme Occupation Measures in Markov Decision Processes with an Absorbing State Open
.In this paper, we consider a Markov decision process (MDP) with a Borel state space \(\textbf{X}\cup \{\Delta \}\), where \(\Delta\) is an absorbing state (cemetery), and a Borel action space \(\textbf{A}\). We consider the space of finit…
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Continuous reducibility: functions versus relations Open
It is proved that the Tang-Pequignot reducibility (or reducibility by relatively continuous relations) on a second countable, T0 space X either coincides with the Wadge reducibility for the given topology, or there is no topology on X that…
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Measure and integration on Boolean algebras of regular open subsets in a topological space Open
The regular open subsets of a topological space form a Boolean algebra, where the `join' of two regular open sets is the interior of the closure of their union. A `credence' is a finitely additive probability measure on this Boolean algebr…
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On existence of the support of a Borel measure Open
We present arguments showing that the standard notion of the support of a probabilistic Borel measure is not well defined in every topological space. Our goal is to create a “very inseparable” space and to show the existence of a family of…
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Borel canonization of analytic sets with Borel sections Open
Kanovei, Sabok and Zapletal asked whether every proper -ideal satisfies the following property: given an analytic equivalence relation with Borel classes, there exists a set which is Borel and -positive such that is Borel. We propose a …
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The interplay of classes of algorithmically random objects Open
We study algorithmically random closed subsets of $\cs$\!, algorithmically random continuous functions from $\cs$ to $\cs$\!, and algorithmically random Borel probability measures on $\cs$, especially the interplay between these three clas…
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Invariant Finitely Additive Measures for General Markov Chains and the Doeblin Condition Open
In this paper, we consider general Markov chains with discrete time in an arbitrary measurable (phase) space. Markov chains are given by a classical transition function that generates a pair of conjugate linear Markov operators in a Banach…
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Non-Archimedean TSI Polish groups and their potential Borel complexity spectrum Open
A variation of the Scott analysis of countable structures is applied to actions of non-Archimedean TSI Polish groups acting continuously on a Polish spaces. We give results on the potential Borel complexity spectrum of such groups, and def…
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Torsion-Free Abelian Groups are Borel Complete Open
We prove that the Borel space of torsion-free Abelian groups with domain $ω$ is Borel complete, i.e., the isomorphism relation on this Borel space is as complicated as possible, as an isomorphism relation. This solves a long-standing open …