Compactivorous sets in Banach spaces

Abstract

A set E E in a Banach space X X is compactivorous if for every compact set K K in X X there is a nonempty, (relatively) open subset of K K which can be translated into E E . In a separable Banach space, this is a sufficient condition which guarantees the Haar nonnegligibility of Borel subsets. We give some characterisations of this property in both separable and nonseparable Banach spaces and prove an extension of the main theorem to countable products of locally compact Polish groups.

Weil-Nachbin Theory for Locally Compact Groups

“…The measure theory measure on locally compact topological spaces is a dynamically developing branch of mathematics, a few new articles pertaining to this topic can be found in the list of references [1][2][3][4][5][6][7][8][9][10][11][12].…”

Haar null closed and convex sets in separable Banach spaces

“…The idea for what follows next comes from a well‐known characterisation of compactness in Banach spaces due to Grothendiek, which is commonly referred to as the Grothendiek compactness principle: a closed subset $$K$$ of a Banach space $$X$$ is compact if and only if there is a sequence $$\mathbf {x}\in c_0(X)$$ such that $$K\subseteq \overline{\mbox{conv}}(\mathbf {x})$$. The Grothendiek compactness principle will turn out to be useful in combination with the main result of [9] (Theorem 4.3), which provides a characterisation of compactivorous sets in Banach spaces. We are especially interested in the second assertion of the theorem.…”