Haar-smallest sets

Kwela, Adam

doi:10.1016/j.topol.2019.106892

Cited by 5 publications

(9 citation statements)

References 19 publications

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“…Remark 7.3. Answering a question posed in a preceding version of this paper, Adam Kwela [21] constructed two compact null-finite subsets A, B on the real line, whose union A ∪ B is not null-finite. This means that the family of subsets of (closed) Borel null-finite subsets on the real line is not an ideal, and the ideal σNF R contains compact subsets of the real line, which fail to be null-finite.…”

Section: The σ-Ideal Generated By (Closed) Borel Null-finite Setsmentioning

confidence: 99%

Null-finite sets in topological groups and their applications

Банах

Jabłońska

2019

Isr. J. Math.

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In the paper we introduce and study a new family of "small" sets which is tightly connected with two well known σ-ideals: of Haar-null sets and of Haar-meager sets. We define a subset A of a topological group X to be null-finite if there exists a convergent sequence (xn)n∈ω in X such that for every x ∈ X the set {n ∈ ω : x + xn ∈ A} is finite. We prove that each null-finite Borel set in a complete metric Abelian group is Haar-null and Haar-meager. The Borel restriction in the above result is essential as each non-discrete metric Abelian group is the union of two null-finite sets. Applying null-finite sets to the theory of functional equations and inequalities, we prove that a mid-point convex function f : G → R defined on an open convex subset G of a metric linear space X is continuous if it is upper bounded on a subset B which is not null-finite and whose closure is contained in G. This gives an alternative short proof of a known generalization of Bernstein-Doetsch theorem (saying that a mid-point convex function f : G → R defined on an open convex subset G of a metric linear space X is continuous if it is upper bounded on a non-empty open subset B of G). Since Borel Haar-finite sets are Haar-meager and Haar-null, we conclude that a mid-point convex function f : G → R defined on an open convex subset G of a complete linear metric space X is continuous if it is upper bounded on a Borel subset B ⊂ G which is not Haar-null or not Haar-meager in X. The last result resolves an old problem in the theory of functional equations and inequalities posed by Baron and Ger in 1983.2

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Section: The σ-Ideal Generated By (Closed) Borel Null-finite Setsmentioning

confidence: 99%

Null-finite sets in topological groups and their applications

Банах

Jabłońska

2019

Isr. J. Math.

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“…Subsets of Polish groups which are not null-finite are known in the literature as subuniversal sets of Kestleman [38], [10] or null-shift-precompact sets of Bingham and Ostaszewski [11], [12]. Haar-finite and Haar-n sets in the real line were explored by Kwela [39].…”

Section: Null-finite Haar-finite and Haar-countable Setsmentioning

confidence: 99%

“…Remark 10.17. In [39] Kwela constructed two Haar-finite compact subsets of R whose union is not null-finite in the real line. Kwela also constructed an example of a compact subset of the real line which is Haar-finite but not Haar-n for every n ∈ N.…”

Section: Null-finite Haar-finite and Haar-countable Setsmentioning

confidence: 99%

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Haar-$\mathcal I$ sets: looking at small sets in Polish groups through compact glasses

Банах

Głąb

Jabłońska

et al. 2021

Dissertationes Math.

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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-I set in a Polish group. Here I is an ideal of subsets of some compact metrizable space K. A Borel subset B ⊆ X of a Polish group X is called Haar-I if there exists a continuous map f : K → X such that f −1 (B + x) ∈ I for all x ∈ X. Moreover, B is generically Haar-I if the set of witness functions {f ∈ C(K, X) : ∀x ∈ X f −1 (B + x) ∈ I} is comeager in the function space C(K, X). We study (generically) Haar-I sets in Polish groups for many concrete and abstract ideals I, and construct the corresponding distinguishing examples. We prove some results on Borel hulls of Haar-I sets, generalizing results of Solecki, Elekes, Vidnyánszky, Doležal, Vlasǎk on Borel hulls of Haar-null and Haar-meager sets. We also establish various Steinhaus properties of the families of (generically) Haar-I sets in Polish groups for various ideals I.Acknowledgements. We would like to express our sincere thanks to Sławomir Solecki for clarifying the situation with the unpublished result of Hjorth [34] (on the Σ 1 1 -hardness of the family CND of closed non-dominating sets) and sending us a copy of the handwritten notes of Hjorth.Special thanks are due to the anonymous referee for careful reading of the manuscript and many valuable remarks and suggestions.

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“…[5]), the σ-ideal of countable sets and Haar-countable sets, the ideal of finite sets and Haar-finite sets and the semi-ideal of sets of cardinality at most n and Haar-n sets (cf. [13,Proposition 1.2]).…”

Section: Clearlymentioning

confidence: 99%

Differentiability of continuous functions in terms of Haar-smallness

Kwela¹,

Wołoszyn²

2018

Preprint

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One of the classical results concerning differentiability of continuous functions states that the set SD of somewhere differentiable functions (i.e., functions which are differentiable at some point) is Haar-null in the space C[0, 1]. By a recent result of Banakh et al., a set is Haar-null provided that there is a Borel hull B ⊇ A and a continuous mapWe prove that SD is not Haar-countable (i.e., does not satisfy the above property with "Lebesgue's null" replaced by "countable", or, equivalently, for each copy C of {0, 1} N there is an h ∈ C[0, 1] such that SD ∩ (C + h) is uncountable.Moreover, we use the above notions in further studies of differentiability of continuous functions. Namely, we consider functions differentiable on a set of positive Lebesgue's measure and functions differentiable almost everywhere with respect to Lebesgue's measure. Furthermore, we study multidimensional case, i.e., differentiability of continuous functions defined on [0, 1] k . Finally, we pose an open question concerning Takagi's function.

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Haar-smallest sets

Cited by 5 publications

References 19 publications

Null-finite sets in topological groups and their applications

Null-finite sets in topological groups and their applications

Haar-$\mathcal I$ sets: looking at small sets in Polish groups through compact glasses

Differentiability of continuous functions in terms of Haar-smallness

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