Inscribing nonmeasurable sets

Measurability Properties of Sets of Vitali's Type

Assume a group G acts on a set. Given a subgroup H of G , by an //-selector we mean a selector of the set of all orbits of H . We investigate measurability properties of //-selectors with respect to G-invariant measures. Let us fix a set A and a group G acting on it. By p we denote a G-invariant countably additive measure on A . The most common example of such a situation is an invariant measur...

On Inscribing «-dimensional Sets in a Regular »-simplex

The Nonmeasurability of Bernstein Sets and Related Topics

In this article, we shall explore the constructions of Bernstein sets, and prove that every Bernstein set is nonmeasurable and doesn’t have the property of Baire. We shall also prove that Bernstein sets don’t have the perfect set property.

On Vector Sums of Measure Zero Sets

We consider the behaviour of measure zero subsets of a vector space under the operation of vector sum. The question whether the vector sum of such sets can be nonmeasurable is discussed in connection with the measure extension problem, and a certain generalization of the classical Sierpiński result [3] is presented. 2000 Mathematics Subject Classification: 28A05, 28D05.

On Measurable Sierpiński-zygmund Functions

It is proved that there exists a Sierpiński-Zygmund function, which is measurable with respect to a certain invariant extension of the Lebesgue measure on the real line R. Let E be a nonempty set and let f : E → R be a function. We say that f is absolutely nonmeasurable if f is nonmeasurable with respect to any nonzero σ-finite diffused (i.e., continuous) measure μ defined on a σ-algebra of sub...