Meager Sets Research Articles

A base-matrix lemma for sets of rationals modulo nowhere dense sets

We study some properties of the quotient forcing notions $${Q_{tr(I)} = \wp(2^{< \omega})/tr(I)}$$ and P I = B(2 ? )/I in two special cases: when I is the ?-ideal of meager sets or the ?-ideal of null sets on 2 ? . We show that the remainder forcing R I = Q tr(I)/P I is ?-closed in these cases. We also study the cardinal invariant of the continuum $${\mathfrak{h}_{\mathbb{Q}}}$$ , the distributivity number of the quotient $${Dense(\mathbb{Q})/nwd}$$ , in order to show that $${\wp(\mathbb{Q})/nwd}$$ collapses $${\mathfrak{c}}$$ to $${\mathfrak{h}_{\mathbb{Q}}}$$ , thus answering a question addressed in Balcar et al. (Fundamenta Mathematicae 183:59---80, 2004).

On the families of sets without the Baire property generated by the Vitali sets

Let A be the family of all meager sets of the real line ℝ, V be the family of all Vitali sets of ℝ, V1be the family of all finite unions of elements of V and V2 = {(C \ A1) ∪ A2: C ∈ V1; A1, A2 ∈ A ...

On countable unions of nonmeager sets in hereditarily Lindelöf spaces

It is well known that any Vitali set on the real line ℝ does not possess the Baire property. The same is valid for finite unions of Vitali sets. What can be said about infinite unions of Vitali sets? Let S be a Vitali set, Sr be the image of S under the translation of ℝ by a rational number r and F = {Sr: r is rational}. We prove that for each non-empty proper subfamily F′ of F the union ∪F′ does not possess the Baire property. We say that a subset A of ℝ possesses Vitali property if there exist a non-empty open set O and a meager set M such that A ⊃ O \ M. Then we characterize those non-empty proper subfamilies F′ of F which unions ∪F′ possess the Vitali property.

Inscribing nonmeasurable sets

Our main inspiration is the work in paper (Gitik and Shelah in Isr J Math 124(1):221---242, 2001). We will prove that for a partition $${\mathcal{A}}$$ of the real line into meager sets and for any sequence $${\mathcal{A}_n}$$ of subsets of $${\mathcal{A}}$$ one can find a sequence $${\mathcal{B}_n}$$ such that $${\mathcal{B}_{n}}$$ 's are pairwise disjoint and have the same "outer measure with respect to the ideal of meager sets". We get also generalization of this result to a class of ?-ideals posessing Suslin property. However, in that case we use additional set-theoretical assumption about non-existing of quasi-measurable cardinal below continuum.

Concerning some results of Pettis

Pettis in [5] obtained improved versions of some theorems of Mc-Shane [4]. Pettis exploited the properties of meager or first category sets throughout his paper. As second category sets in a topological space of second category form a grill, an alternative approach to the results of Pettis via grills may well be conjectured. Indeed, such a grill and even arbitrary grills satisfying requisite conditions are employed in this article to achieve certain extended forms of some results of Pettis.

The number of unit distances is almost linear for most norms

We prove that there exists a norm in the plane under which no n-point set determines more than O ( n log n log log n ) unit distances. Actually, most norms have this property, in the sense that their complement is a meager set in the metric space of all norms (with the metric given by the Hausdorff distance of the unit balls).

On the convergence of Fourier series of computable Lebesgue integrable functions

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of Lp-computable functions (computable Lebesgue integrable functions) with a size notion, by introducing Lp-computable Baire categories. We show that Lp-computable Baire categories satisfy the following three basic properties. Singleton sets {f } (where f is Lp-computable) are meager, suitable infinite unions of meager sets are meager, and the whole space of Lp-computable functions is not meager. We give an alternative characterization of meager sets via Banach-Mazur games. We study the convergence of Fourier series for Lp-computable functions and show that whereas for every p > 1, the Fourier series of every Lp-computable function f converges to f in the Lp norm, the set of L1-computable functions whose Fourier series does not diverge almost everywhere is meager (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Extending Baire property by uncountably many sets

AbstractWe show that for an uncountable κ in a suitable Cohen real model for any family {Av}v&lt;κ of sets of reals there is a σ-homomorphism h from the σ-algebra generated by Borel sets and the sets Av, into the algebra of Baire subsets of 2κ modulo meager sets such that for all Borel B,The proof is uniform, works also for random reals and the Lebesgue measure, and in this way generalizes previous results of Carlson and Solovay for the Lebesgue measure and of Kamburelis and Zakrzewski for the Baire property.

Infinite combinatorics in function spaces: Category methods

The infinite combinatorics here give statements in which, from some sequence, an infinite subsequence will satisfy some condition - for example, belong to some specified set. Our results give such statements generically - that is, for 'nearly all' points, or as we shall say, for quasi all points - all off a null set in the measure case, or all off a meagre set in the category case. The prototypical result here goes back to Kestelman in 1947 and to Borwein and Ditor in the measure case, and can be extended to the category case also. Our main result is what we call the Category Embedding Theorem, which contains the Kestelman-Borwein-Ditor Theorem as a special case. Our main contribution is to obtain function wise rather than point wise versions of such results. We thus subsume results in a number of recent and related areas, concerning e.g., additive, subadditive, convex and regularly varying functions.

Universally meager sets, II

We present new characterizations of universally meager sets, shown in [P. Zakrzewski, Universally meager sets, Proc. Amer. Math. Soc. 129 (6) (2001) 1793–1798] to be a category analog of universally null sets. In particular, we address the question of how this class is related to another class of universally meager sets, recently introduced by Todorcevic [S. Todorcevic, Universally meager sets and principles of generic continuity and selection in Banach spaces, Adv. Math. 208 (2007) 274–298].

Covering a Polish group by translates of a nowhere dense set

We show that, for every nonlocally compact Polish group G with a left-invariant complete metric ρ, we have cov G = cov ( M ) . Here, cov G is the minimal number of translates of a fixed closed nowhere dense subset of G , which is needed to cover G , and cov ( M ) is the minimal cardinality of a cover of the real line R by meagre sets.

On the Convergence of Fourier Series of Computable Lebesgue Integrable Functions

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of Lp-computable functions (computable Lebesgue integrable functions) with a size notion, by introducing Lp-computable Baire categories. We show that Lp-computable Baire categories satisfy the following three basic properties. Singleton sets {f} (where f is Lp-computable) are meager, suitable infinite unions of meager sets are meager, and the whole space of Lp-computable functions is not meager. We give an alternative characterization of meager sets via Banach Mazur games. We study the convergence of Fourier series for Lp-computable functions and show that whereas for every p>1, the Fourier series of every Lp-computable function f converges to f in the Lp norm, the set of L1-computable functions whose Fourier series does not diverge almost everywhere is meager.

On Lower and Semi-Lower Density Operators

Abstract This paper contains some results connected with topologies generated by lower and semi-lower density operators. We show that in some measurable spaces (𝑋, 𝑆, 𝐽) there exists a semi-lower density operator which does not generate a topology. We investigate some properties of nowhere dense sets, meager sets and σ-algebras of sets having the Baire property, associated with the topology generated by a semi-lower density operator.

A Fubini theorem

Let I 0 be the σ-ideal of subsets of a Polish group generated by Borel sets which have perfectly many pairwise disjoint translates. We prove that a Fubini-type theorem holds between I 0 and the σ-ideals of Haar measure zero sets and of meager sets. We use this result to give a simple proof of a generalization of a theorem of Balcerzak–Rosłanowski–Shelah stating that I 0 on 2 N strongly violates the countable chain condition.

A Fubini theorem

Let I 0 be the σ -ideal of subsets of a Polish group generated by Borel sets which have perfectly many pairwise disjoint translates. We prove that a Fubini-type theorem holds between I 0 and the σ -ideals of Haar measure zero sets and of meager sets. We use this result to give a simple proof of a generalization of a theorem of Balcerzak–Rosłanowski–Shelah stating that I 0 on 2 N strongly violates the countable chain condition.

Classification of cocycles of generic equivalence relations

We study cocycles of an ergodic generic countable equivalence relation ℜ modulo meager sets. Two cocycles of ℜ are called weakly equivalent if they are cohomologous up to an element of Aut ℜ. It is proved that two nontransient cocycles with values in an arbitrary countable group are weakly equivalent if and only if their generic Mackey actions are isomorphic.

Universally meager sets and principles of generic continuity and selection in Banach spaces

In this paper we use large cardinals to address some problems about generic continuity and generic selection that occur in the study of fragmentability and differentiability in the context of Banach spaces.

The number of translates of a closed nowhere dense set required to cover a Polish group

For a Polish group G let cov G be the minimal number of translates of a fixed closed nowhere dense subset of G required to cover G . For many locally compact G this cardinal is known to be consistently larger than cov ( M ) which is the smallest cardinality of a covering of the real line by meagre sets. It is shown that for several non-locally compact groups cov G = cov ( M ) . For example the equality holds for the group of permutations of the integers, the additive group of a separable Banach space with an unconditional basis and the group of homeomorphisms of various compact spaces.

Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures

We consider the question of which of the major classes defined by topological diagonalizations of open or Borel covers is hereditary. Many of the classes in the open case are not hereditary already in ZFC, and none of them are provably hereditary. This is in contrast with the Borel case, where some of the classes are provably hereditary. Two of the examples are counter-examples of sizes d \mathfrak {d} and b \mathfrak {b} , respectively, to the Menger and Hurewicz Conjectures, and one of them answers a question of Steprans on perfectly meager sets.

Category product densities and liftings

In this paper we investigate two main problems. One of them is the question on the existence of category liftings in the product of two topological spaces. We prove, that if X × Y is a Baire space, then, given (strong) category liftings ρ and σ on X and Y, respectively, there exists a (strong) category lifting π on the product space such that π is a product of ρ and σ and satisfies the following section property: [ π ( E ) ] x = σ ( [ π ( E ) ] x ) for all E ⊆ X × Y with Baire property and all x ∈ X . We give also an example, where some of the sections [ π ( E ) ] y must be without Baire property. Then, we investigate the existence of densities respecting coordinates on products of topological spaces, provided these products are Baire spaces. The densities are defined on σ-algebras of sets with Baire property and select elements modulo the σ-ideal of all meager sets. In all the problems the situation in the “category case” turns out to be much better, than in case of products of measure spaces. In particular, in every product there exists a canonical strong density being a product of the canonical densities in the factors and there exist (strong) densities respecting coordinates with given a priori marginal (strong) densities.