Jakub Duda

The Banach-Zarecki theorem for functions with values in metric spaces

Using an old result of Luzin about his property (N), we prove a general version of the Banach-Zarecki theorem (on absolute continuity and Luzins property (N)).

On the size of the set of points where the metric projection exists

In this paper we answer in the negative a question due to J. P. R. Christensen about almost everywhere existence of nearest points using a decomposition of ℓ2 due to J. Matoušek and E. Matoušková. We also formulate a similar question about almost everywhere existence of farthest points and answer it in the negative.

Null sets and games in Banach spaces

Abstract The notion of Aronszajn-null sets generalizes the notion of Lebesgue measure zero in the Euclidean space to infinite dimensional Banach spaces. We present a game-theoretic approach to Aronszajn-null sets, establish its basic properties, and discuss some ensuing open problems.