Let G be an abelian Polish group, e.g. a separable Banach space. A subset X ⊂ G is called Haar null (in the sense of Christensen) if there exists a Borel set B ⊃ X and a Borel probability measure μ on G such that μ(B+g) = 0 for every g ∈ G. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent. Answering an old question of Mycielski we show that if G...

In this paper we further a previous study concerning abstract regularity for monotone set multifunctions, with has immediate applications in well-known situations such as the Borel σ-algebra of a Hausdorff space and/or the Borel (Baire, respectively) δ-ring or σ-ring of a locally compact Hausdorff space. We also study relationships among abstract regularities and other properties of continuity....

In this note an extension-type theorem for null-additive fuzzy measures is proved. AMS Mathematics Subject Classification (2000): 28A25

That is, a smooth one-parameter family of measurable hypersurfaces may be translated to lie in a null set. Moreover, the translations may be taken parallel to R and need not depend on g. Our motivation came from studying curved analogues of the Kakeya and Nikodym problems: we wished to show that a null set could contain a translate of every member of a specified family of curves, or in the Niko...

This article describes how a mixture of two densities, f0 and f1, may be decomposed into a different mixture consisting of three densities. These new densities, f+, f−, and f=, summarize differences between f0 and f1: f+ is high in areas of excess of f1 compared to f0; f− represents deficiency of f1 compared to f0 in the same way; f= represents commonality between f1 and f0. The supports of f+ ...

In this paper, we study different types of non-additive set multifunctions (such as: uniformly autocontinuous, null-additive, null-null-additive), presenting relationships among them and some of their properties regarding atoms and pseudo-atoms. We also study non-atomicity and non-pseudo-atomicity of regular null-additive set multifunctions defined on the Baire (Borel respectively) δ-ring of a ...