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Tamás Keleti - One of the best experts on this subject based on the ideXlab platform.
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decomposing the real line into Borel Sets closed under addition
Mathematical Logic Quarterly, 2015Co-Authors: Márton Elekes, Tamás KeletiAbstract:We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in and even in the theory if the number of pieces can be uncountable but less than the continuum. We also investigate various versions: what happens if we drop the Borelness requirement, if we replace addition by multiplication, if the pieces are subgroups, if we partition (0, ∞), and so on.
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decomposing the real line into Borel Sets closed under addition
arXiv: Logic, 2014Co-Authors: Márton Elekes, Tamás KeletiAbstract:We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in $ZFC$ and even in the theory $ZFC + \mathfrak{c} = \omega_2$ if the number of pieces can be uncountable but less than the continuum. We also investigate various versions: what happens if we drop the Borelness requirement, if we replace addition by multiplication, if the pieces are subgroups, if we partition $(0,\infty)$, and so on.
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Borel Sets which are null or non-$\sigma$-finite for every translation invariant measure
arXiv: Classical Analysis and ODEs, 2011Co-Authors: Márton Elekes, Tamás KeletiAbstract:We show that the set of Liouville numbers is either null or non-$\sigma$-finite with respect to every translation invariant Borel measure on $\RR$, in particular, with respect to every Hausdorff measure $\iH^g$ with gauge function $g$. This answers a question of D. Mauldin. We also show that some other simply defined Borel Sets like non-normal or some Besicovitch-Eggleston numbers, as well as all Borel subgroups of $\RR$ that are not $F_\sigma$ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such measure on it can be arbitrary.
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is lebesgue measure the only sigma finite invariant Borel measure
arXiv: Classical Analysis and ODEs, 2011Co-Authors: Márton Elekes, Tamás KeletiAbstract:R.D.Mauldin asked if every translation invariant $\sigma$-finite Borel measure on $\RR^d$ is a constant multiple of Lebesgue measure. The aim of this paper is to show that the answer is "yes and no", since surprisingly the answer depends on what we mean by Borel measure and by constant. We present Mauldin's proof of what he called a folklore result, stating that if the measure is only defined for Borel Sets then the answer is affirmative. Then we show that if the measure is defined on a $\sigma$-algebra \emph{containing} the Borel Sets then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant $\sigma$-finite Borel measure (in the wider sense) on $\RR^d$ can be non-$\sigma$-finite when we restrict it to the Borel Sets.
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Borel Sets which are null or non- σ -finite for every translation invariant measure
Advances in Mathematics, 2006Co-Authors: Márton Elekes, Tamás KeletiAbstract:Abstract We show that the set of Liouville numbers is either null or non- σ -finite with respect to every translation invariant Borel measure on R , in particular, with respect to every Hausdorff measure H g with gauge function g . This answers a question of R.D. Mauldin. We also show that some other simply defined Borel Sets like non-normal or some Besicovitch–Eggleston numbers, as well as all Borel subgroups of R that are not F σ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such measure on it can be arbitrary.
Márton Elekes - One of the best experts on this subject based on the ideXlab platform.
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decomposing the real line into Borel Sets closed under addition
Mathematical Logic Quarterly, 2015Co-Authors: Márton Elekes, Tamás KeletiAbstract:We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in and even in the theory if the number of pieces can be uncountable but less than the continuum. We also investigate various versions: what happens if we drop the Borelness requirement, if we replace addition by multiplication, if the pieces are subgroups, if we partition (0, ∞), and so on.
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decomposing the real line into Borel Sets closed under addition
arXiv: Logic, 2014Co-Authors: Márton Elekes, Tamás KeletiAbstract:We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in $ZFC$ and even in the theory $ZFC + \mathfrak{c} = \omega_2$ if the number of pieces can be uncountable but less than the continuum. We also investigate various versions: what happens if we drop the Borelness requirement, if we replace addition by multiplication, if the pieces are subgroups, if we partition $(0,\infty)$, and so on.
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Borel Sets which are null or non-$\sigma$-finite for every translation invariant measure
arXiv: Classical Analysis and ODEs, 2011Co-Authors: Márton Elekes, Tamás KeletiAbstract:We show that the set of Liouville numbers is either null or non-$\sigma$-finite with respect to every translation invariant Borel measure on $\RR$, in particular, with respect to every Hausdorff measure $\iH^g$ with gauge function $g$. This answers a question of D. Mauldin. We also show that some other simply defined Borel Sets like non-normal or some Besicovitch-Eggleston numbers, as well as all Borel subgroups of $\RR$ that are not $F_\sigma$ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such measure on it can be arbitrary.
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is lebesgue measure the only sigma finite invariant Borel measure
arXiv: Classical Analysis and ODEs, 2011Co-Authors: Márton Elekes, Tamás KeletiAbstract:R.D.Mauldin asked if every translation invariant $\sigma$-finite Borel measure on $\RR^d$ is a constant multiple of Lebesgue measure. The aim of this paper is to show that the answer is "yes and no", since surprisingly the answer depends on what we mean by Borel measure and by constant. We present Mauldin's proof of what he called a folklore result, stating that if the measure is only defined for Borel Sets then the answer is affirmative. Then we show that if the measure is defined on a $\sigma$-algebra \emph{containing} the Borel Sets then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant $\sigma$-finite Borel measure (in the wider sense) on $\RR^d$ can be non-$\sigma$-finite when we restrict it to the Borel Sets.
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Borel Sets which are null or non- σ -finite for every translation invariant measure
Advances in Mathematics, 2006Co-Authors: Márton Elekes, Tamás KeletiAbstract:Abstract We show that the set of Liouville numbers is either null or non- σ -finite with respect to every translation invariant Borel measure on R , in particular, with respect to every Hausdorff measure H g with gauge function g . This answers a question of R.D. Mauldin. We also show that some other simply defined Borel Sets like non-normal or some Besicovitch–Eggleston numbers, as well as all Borel subgroups of R that are not F σ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such measure on it can be arbitrary.
Olivier Finkel - One of the best experts on this subject based on the ideXlab platform.
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Topological Properties of Omega Context Free Languages
2015Co-Authors: Olivier FinkelAbstract:This paper is a study of topological properties of omega context free languages (ω-CFL). We first extend some decidability results for the deterministic ones (ω-DCFL), proving that one can decide whether an ω-DCFL is in a given Borel class, or in the Wadge class of a given ω-regular language. We prove that ω-CFL exhaust the hierarchy of Borel Sets of finite rank, and that one cannot decide the Borel class of an ω-CFL, giving an answer to a question of [LT94]. We give also a (partial) answer to a question of [Sim92] about omega powers of finitary languages. We show that Büchi-Landweber’s Theorem cannot be extended to even closed ω-CFL: in a Gale-Stewart game with a (closed) ω-CFL winning set, one cannot decide which player has a winning strategy. From the proof of topological properties we derive some arithmetical properties of ω-CFL
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On Omega Context Free Languages which are Borel Sets of Infinite Rank
arXiv: Logic in Computer Science, 2010Co-Authors: Olivier FinkelAbstract:This paper is a continuation of the study of topological properties of omega context free languages (omega-CFL). We proved before that the class of omega-CFL exhausts the hierarchy of Borel Sets of finite rank, and that there exist some omega-CFL which are analytic but non Borel Sets. We prove here that there exist some omega context free languages which are Borel Sets of infinite (but not finite) rank, giving additional answer to questions of Lescow and Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", Springer LNCS 803 (1994), 583-621].
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On Infinitary Rational Relations and Borel Sets
arXiv: Logic in Computer Science, 2010Co-Authors: Olivier FinkelAbstract:We prove in this paper that there exists some infinitary rational relations which are Sigma^0_3-complete Borel Sets and some others which are Pi^0_3-complete. This implies that there exists some infinitary rational relations which are Delta^0_4-Sets but not (Sigma^0_3U Pi^0_3)-Sets. These results give additional answers to questions of Simonnet and of Lescow and Thomas.
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on infinitary rational relations and Borel Sets
Discrete Mathematics & Theoretical Computer Science, 2003Co-Authors: Olivier FinkelAbstract:We prove in this paper that there exists some infinitary rational relations which are Σ30-complete Borel Sets and some others which are Π30-complete. These results give additional answers to questions of Simonnet [Sim92] and of Lescow and Thomas [Tho90,LT94].
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On omega context free languages which are Borel Sets of infinite rank
Theoretical Computer Science, 2002Co-Authors: Olivier FinkelAbstract:This paper is a continuation of the study of topological properties of omega context free languages (?-CFL). We proved in (Topological properties of omega context free languages, Theoretical Computer Science, 262 (1?2) (2001) 669?697) that the class of ?-CFL exhausts the finite ranks of the Borel hierarchy, and in (Borel hierarchy and omega context free languages, Theoretical Computer Science, to appear) that there exist some ?-CFL which are analytic but non Borel Sets. We prove here that there exist some omega context free languages which are Borel Sets of infinite (but not finite) rank, giving additional answer to questions of Lescow and Thomas Logical specifications of infinite computations in: “A Decade of Concurrency” (J.W. de Bakker et al. (Eds.), Springer LNCS 803 (1994) 583?621).
Matthieu Fradelizi - One of the best experts on this subject based on the ideXlab platform.
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Concentration inequalities for s-concave measures of dilations of Borel Sets and applications
Electronic Journal of Probability, 2009Co-Authors: Matthieu FradeliziAbstract:We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel Sets in the Euclidean space by a s-concave probability measure. Our result gives a common generalization of an inequality of Nazarov, Sodin and Volberg and a concentration inequality of Guedon. Applying our inequality to the level Sets of functions satisfying a Remez type inequality, we deduce, as it is classical, that these functions enjoy dimension free distribution inequalities and Kahane-Khintchine type inequalities with positive and negative exponent, with respect to an arbitrary s-concave probability measure.
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Concentration inequalities for $s$-concave measures of dilations of Borel Sets and applications
arXiv: Probability, 2008Co-Authors: Matthieu FradeliziAbstract:We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel Sets in $\mathbb{R}^n$ by a $s$-concave probability. Our result gives a common generalization of an inequality of Nazarov, Sodin and Volberg and a concentration inequality of Gu\'edon. Applying our inequality to the level Sets of functions satisfying a Remez type inequality, we deduce, as it is classical, that these functions enjoy dimension free distribution inequalities and Kahane-Khintchine type inequalities with positive and negative exponent, with respect to an arbitrary $s$-concave probability.
Alexander S. Kechris - One of the best experts on this subject based on the ideXlab platform.
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Approximation of analytic by Borel Sets and definable countable chain conditions
Israel Journal of Mathematics, 1995Co-Authors: Alexander S. Kechris, Sławomir SoleckiAbstract:Let I be a σ-ideal on a Polish space such that each set from I is contained in a Borel set from I . We say that I fails to fulfil the Σ _ 1 ^ 1 countable chain condition if there is a Σ _ 1 ^ 1 equivalence relation with uncountably many equivalence classes none of which is in I . Assuming definable determinacy, we show that if the family of Borel Sets from I is definable in the codes of Borel Sets, then each Σ _ 1 ^ 1 set is equal to a Borel set modulo a set from I iff I fulfils the Σ _ 1 ^ 1 countable chain condition. Further we characterize the σ-ideals I generated by closed Sets that satisfy the countable chain condition or, equivalently in this case, the approximation property for Σ _ 1 ^ 1 Sets mentioned above. It turns out that they are exactly of the form MGR ( F )={ A : ∀ F ∈ F A ∩ F is meager in F } for a countable family F of closed Sets. In particular, we verify partially a conjecture of Kunen by showing that the σ-ideal of meager Sets is the unique σ-ideal on R , or any Polish group, generated by closed Sets which is invariant under translations and satisfies the countable chain condition.
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Borel Sets and Baire Category
Graduate Texts in Mathematics, 1995Co-Authors: Alexander S. KechrisAbstract:Every Borel sot has the BP, and every Borel function is Baire measurable. We will calculate next the complexity of the property of being meager for Borel Sets.
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Borel Sets and Measures
Graduate Texts in Mathematics, 1995Co-Authors: Alexander S. KechrisAbstract:Let (X,S) be a measurable space. A measure on (X,S) is a map µ: S → [0,∞] such that µ (O) = 0 and \(\mu \left( {\bigcup\nolimits_n {{A_n}} } \right) = {\Sigma _n}\mu \left( {{A_n}} \right)\)for any pairwise disjoint family {An} \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \subset } \)S. A measure space is a triple (X,S,µ), where (X,S) is a measurable space and µ is a measure on (X,S). We often write (X,µ) when there is no danger of confusion.
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Borel Sets and Functions
Graduate Texts in Mathematics, 1995Co-Authors: Alexander S. KechrisAbstract:Let (X,T) be a topological space. The class of Borel Sets of X is the σ-algebra generated by the open Sets of X. We denote it by B(X,T) (or by B(X) or B(T), when appropriate). We call (X, B(X)) the Borel space of X. If Ɛ is a countable subbasis for X, then clearly B(X) = σ(Ɛ), so B(X) is countably generated when X is second countable. Note also that if Y is a subspace of X then (Y, B(Y)) is a subspace of (X,B(X)) (i.e., B(Y) = B(X)|Y). It is obvious that B(X) contains all open, closed, Fσ, and Gδ Sets in X.