Remarks on nonmeasurable unions of big point families

We show that under some conditions on a family A ? I there exists a subfamily A₀ ? A such that ∪ A₀ is nonmeasurable with respect to a fixed ideal I with Borel base of a fixed uncountable Polish space. Our result applies to the classical ideal of null subsets of the real line and to the ideal of first category subsets of the real line (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On nonmeasurable unions

For a Polish space X and a σ-ideal I of subsets of X which has a Borel base we consider families A of sets in I with the union ?A not in I. We determine several conditions on A which imply the existence of a subfamily A^′ of A whose union ?A^′ is not in the σ-field generated by the Borel sets on X and I. Main examples are X=R and I being the ideal of sets of Lebesgue measure zero or the ideal of sets of the first Baire category.     

Hamel‐isomorphic images of the unit ball

In this article we consider linear isomorphisms over the field of rational numbers between the linear spaces ?² and ?. We prove that if f is such an isomorphism, then the image by f of the unit disk is a strictly nonmeasurable subset of the real line, which has different properties than classical non‐measurable subsets of reals. We shall also consider the question whether all images of bounded measurable subsets of the plane via a such mapping are non‐measurable (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Uncountable Unions and Intersections of Measurable Sets

We consider several natural situations where the union or intersection of an uncountable family of measurable (in various senses) sets with a good additional structure is again measurable or may fail to be measurable. We primarily deal with Lebesgue measurable sets and sets with the Baire property. In particular, uncountable unions of sets homeomorphic to a closed Euclidean simplex are considered in detail, and it is shown that the Lebesgue measure and the Baire property differ essentially in this aspect. Another difference between measure and category is illustrated in the case of some uncountable intersections of sets of full measure (comeager sets, respectively). We also discuss a topological form of the Vitali covering theorem, in connection with the Baire property of uncountable unions of certain sets.     

One-to-one Borel selection theorems

For we obtain new theorems stating that a Borel set in with large sets of large vertical and large horizontal sections admits a one-to-one Borel selection with large domain and large range. Largeness is meant mainly in measure or category sense. Our proofs combine a result of Graf and Mauldin with a modified result of Sarbadhikari.

     

Some results on projection of planar sets

In this paper we define certain types of projections of planar sets and study some properties of such projections.     

On the difference property of the family of functions with the Baire property

We show that it is consistent with ZFC that the family of functions with the Baire property has the difference property. That is, every function for which f(x + h)-f(x) has the Baire property for every h∈R is of the form f=g + Awhere g has the Baire property and A is additive. This revised version was published online in June 2006 with corrections to the Cover Date.     

Small subsets of the reals and tree forcing notions

We discuss the question of which properties of smallness in the sense of measure and category (e.g. being a universally null, perfectly meager or strongly null set) imply the properties of smallness related to some tree forcing notions (e.g. the properties of being Laver-null or Miller-null).

     

Perfectly meager sets and universally null sets

We will show that there is no example of a set distinguishing between universally null and perfectly meager sets.

     

On sets with Baire property in topological spaces

Steinhaus [9] prove that if a set A has a positive Lebesgue measure in the line then its distance set contains an interval. He obtained even stronger forms of this result in [9], which are concerned with mutual distances between points in an infinite sequence of sets. Similar theorems in the case we replace distance by mutual ratio were established by Bose-Majumdar [1]. In the present paper, we endeavour to obtain some results related to sets with Baire property in locally compact topological spaces, particular cases of which yield the Baire category analogues of the above results of Steinhaus [9] and their corresponding form for ratios by Bose-Majumdar [1].     

Strongly meager sets and their uniformly continuous images

We prove the following theorems:

(1) Suppose that is a continuous function and is a Sierpinski set. Then

(A)

for any strongly measure zero set , the image is an -set,

(B)

is a perfectly meager set in the transitive sense.

(2) Every strongly meager set is completely Ramsey null.

     

On translations of subsets of the real line

In this paper we discuss various questions connected with translations of subsets of the real line. Most of these questions originate from W. Sierpinski. We discuss the number of translations a single subset of the reals may have. Later we discuss almost invariant subsets of Abelian groups.

     

关于自伴算子的奇异连续谱

本文建立了自伴算子奇异连续谱的三个定理,它们推广了Barry Simon近期所获得的一些重要结果.     

Choice principles for special subsets of the real line

We study the role the axiom of choice plays in the existence of some special subsets of ? and its power set ?(?).     

Some remarks on totally imperfect sets

We prove the following two theorems.

Theorem 1. Let be a strongly meager subset of . Then it is dual Ramsey null.

We will say that a -ideal of subsets of satisfies the condition iff for every , if

then .

Theorem 2. The -ideals of perfectly meager sets, universally meager sets and perfectly meager sets in the transitive sense satisfy the condition .

     

On approximately additive functions

Let C be a convex symmetric subset of a real Banach space F and K be a subgroup of the group (F,+). Let E be a real linear space, h:E→F, and h(x+y)−h(x)−h(y)∈K+C for x,y∈E. We prove that under some additional assumptions h can be represented in the form: h=A+γ+κ with an additive (or linear) A:E→F and some γ:E→C, κ:E→K.     

On a fragment of the universal Baire property for sets

There is a well-known global equivalence between sets having the universal Baire property, two-step generic absoluteness, and the closure of the universe under the sharp operation. In this note, we determine the exact consistency strength of sets being -cc-universally Baire, which is below . In a model obtained, there is a set which is weakly -universally Baire but not -universally Baire.

     

处处不连续又不可测的达布函数类的势

该文提出了一种构造处处不连续，而且在任何区间内取到任一函数值ｃ次的达布函数类的新方法，证明了该函数类的势为２＾ｃ（ｃ为连续统势）；还得到了处处不连续又不可测的达布函数类的势为２＾ｃ。     

On perfectly meager sets

We show that it is consistent that the product of perfectly meager sets is perfectly meager.