Haar Null Closed and Convex Sets in Separable Banach Spaces

TL;DR

This paper proves that in separable Banach spaces, closed convex sets are Haar null if and only if they are Haar meagre, linking measure and category concepts and applying this to Banach lattice structures.

Contribution

It establishes the equivalence of Haar null and Haar meagre for closed convex sets in separable Banach spaces, and applies this to solve existing conjectures and characterize Banach lattices.

Findings

Haar null and Haar meagre sets coincide for closed convex sets in separable Banach spaces

Resolved a conjecture by Esterle, Matheron, and Moreau

Characterized Banach lattices with non-Haar null positive cones

Abstract

Haar null sets were introduced by J.P.R. Christensen in 1972 to extend the notion of sets with zero Haar measure to nonlocally compact Polish groups. In 2013, U.B. Darij defined a categorical version of Haar null sets, which he named Haar meagre sets. The present paper aims to show that, whenever $C$ is a closed, convex subset of a separable Banach space, $C$ is Haar null if and only if $C$ is Haar meagre. We then use this fact to improve a theorem of E. Matou\v{s}kov\'{a} and to solve a conjecture proposed by Esterle, Matheron and Moreau. Finally, we apply the main theorem to find a characterisation of separable Banach lattices whose positive cone is not Haar null.