Descriptive set Theory of Families of Small Sets

Matheron, Étienne; Zelený, Miroslav

doi:10.2178/bsl/1203350880

Cited by 13 publications

(7 citation statements)

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“…As a warm up, we calculate the complexity of the codes of Haar-null analytic subsets of and the complexity of the closed Haar null subsets of Z ω in the Effros Borel space. It has been shown by Solecki [13], see also [10,15], that the codes for the closed Haar-null subsets, as well as the set {C ∈ F (Z ω ) : C ∈ HN } are neither analytic nor co-analytic. (In fact, Z ω can be replaced by any non-locally compact Polish group admitting a two-sided invariant metric).…”

Section: Complexity Estimationmentioning

confidence: 99%

Haar-positive closed subsets of Haar-positive analytic sets

Elekes¹,

Poór²,

Vidnyánszky³

2020

Preprint

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We show that every non-Haar-null analytic subset of Z ω contains a non-Haar-null closed subset. Moreover, we also prove that the codes of Haar-null analytic subsets, and, consequently, closed Haar-null sets in the Effros Borel space of Z ω form a ∆ 1 2 set.It is not hard to see that non-locally-compact Polish groups do not admit a Haar measure (that is, an invariant σ-finite Borel measure). However, Christensen [4] (and later, independently, Hunt-Sauer-Yorke [8]) generalized the ideal of Haar measure zero sets to every Polish group as follows: Definition 0.1. Let (G, •) be a Polish group and S ⊂ G. We say that S is Haar-null, (in symbols, S ∈ HN ) if there exists a universally measurable set U ⊃ S (that is, a set measurable with respect to every Borel probability measure) and a Borel probability measure µ on G such that for every g, h ∈ G we have µ(gUh) = 0. Such a measure µ is called a witness measure for S. This notion has found wide application in diverse areas such as functional analysis, dynamical systems, group theory, geometric measure theory, and analysis (see, e.g., [12,2,14,13,5,1]). It provides a well-behaved notion of "almost every" (or "prevalent") element of a Polish group. It is natural to investigate the regularity properties of Haar-null sets. In particular, one might wonder whether "small sets are contained in nice small sets" and whether "large sets contain nice large sets". Concerning the first question, Solecki [13] has shown a positive statement, namely, that every analytic Haar-null set is contained in a Borel Haar-null set. On the negative side, the first and the third author [6] proved that, unlike the situation in locally-compact groups,

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Section: Complexity Estimationmentioning

confidence: 99%

Haar-positive closed subsets of Haar-positive analytic sets

Elekes¹,

Poór²,

Vidnyánszky³

2020

Preprint

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“…, the sequence n∈ω I n also converges to A, and as J is a σ-ideal, A ∪ n∈ω I n ∈ J . For more information on ideals of compact sets consult [9,11,12,15,19].…”

Section: Lemma 22 ([4]mentioning

confidence: 99%

“…On the other hand, observe that, for each g ∈ ω ω , the set k J k g(k) is the union of a sequence (b i ), where sets b i are chosen from distinct B N n with n, N ∈ ω. Thus, (8) implies that (11) ϕ(…”

Section: A Dichotomy For Flat Idealsmentioning

confidence: 99%

Tukey order among $F_σ$ ideals

He¹,

Hrušák²,

Rojas-Rebolledo³

et al. 2020

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We investigate the Tukey order in the class of Fσ ideals of subsets of ω. We show that no nontrivial Fσ ideal is Tukey below a G δ σ-ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using Tukey reductions and games. For example, we show that all gradually flat ideals are Tukey below the density zero ideal Z 0 .

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“…An ideal I is a σ-ideal if it is also closed under countable unions whenever the union itself is compact. Ideals of compact sets arise commonly in analysis out of various notions of smallness; see [3] for a survey of results and applications.After [4] we say that an ideal I has property ( * ) if, for any sequence of sets K n ∈ I, there exists a G δ set G such that n K n ⊆ G and K(G) ⊆ I. Property ( * ) holds in a broad class of G δ ideals that includes all natural examples, including the ideals of compact meager sets, measure-zero sets, sets of dimension ≤ n for fixed n ∈ N, and Z-sets.…”

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confidence: 99%

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confidence: 99%

A note on $G_δ$ ideals of compact sets

Saran

2018

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Solecki has shown that a broad natural class of G δ ideals of compact sets can be represented through the ideal of nowhere dense subsets of a closed subset of the hyperspace of compact sets. In this note we show that the closed subset in this representation can be taken to be closed upwards.Let E be a compact Polish space and let K(E) denote the hyperspace of its compact subsets, equipped with the Vietoris topology. A set I ⊆ K(E) is an ideal of compact sets if it is closed under the operations of taking subsets and finite unions. An ideal I is a σ-ideal if it is also closed under countable unions whenever the union itself is compact. Ideals of compact sets arise commonly in analysis out of various notions of smallness; see [3] for a survey of results and applications.After [4] we say that an ideal I has property ( * ) if, for any sequence of sets K n ∈ I, there exists a G δ set G such that n K n ⊆ G and K(G) ⊆ I. Property ( * ) holds in a broad class of G δ ideals that includes all natural examples, including the ideals of compact meager sets, measure-zero sets, sets of dimension ≤ n for fixed n ∈ N, and Z-sets. (See [4] for these and other examples and a discussion of property ( * ).) Solecki has shown in [4] that ideals in this class are represented via the meager ideal of a closed subset of K(E). The following definition is essential to the representation:Theorem 1 (Solecki). Suppose I is coanalytic and non-empty. Then I has property ( * ) iff there exists a closed set F ⊆ K(E) such that, for anyThis representation is analogous to a result of Choquet [1] that establishes a correspondence between alternating capacities of order ∞ on E and probability Borel measures on K(E).Note that the set F in Theorem 1 is not unique. We hope to determine properties for F that make it a canonical representative, perhaps upto some notion of equivalence. One property of interest is that of being closed upwards, i.e., ∀A, B ∈ K(E), B ⊇ A ∈ F ⇒ B ∈ F. This property ensures that the map K → K * ∩F, a fundamental function in this context, is continuous. In several examples of G δ ideals with property ( * ), the natural choice

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Descriptive set Theory of Families of Small Sets

Abstract: Abstract. This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.

Cited by 13 publications

References 93 publications

Haar-positive closed subsets of Haar-positive analytic sets

Haar-positive closed subsets of Haar-positive analytic sets

Tukey order among $F_σ$ ideals

A note on $G_δ$ ideals of compact sets

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