A Topological Characterisation of Haar Null Convex Sets

TL;DR

This paper characterizes Haar null convex sets in Banach spaces using topological properties of their duals, extending classical measure-zero results to a broader topological framework.

Contribution

It introduces a topological criterion for Haar nullity of convex sets in Banach spaces based on the interior of their weak* closures.

Findings

Haar null convex sets have empty interior in their weak* closure.

Converging sequences of Haar null sets have Haar null limits.

Extension of measure-zero properties to topological characterizations.

Abstract

In $\mathbb{R}^d$, a closed, convex set has zero Lebesgue measure if and only its interior is empty. More generally, in separable, reflexive Banach spaces, closed and convex sets are Haar null if and only if their interior is empty. We extend this facts by showing that a closed, convex set in a separable Banach space is Haar null if and only if its weak$^*$ closure in the second dual has empty interior with respect to the norm topology. It then follows that, in the metric space of all nonempty, closed, convex and bounded subsets of a separable Banach space, converging sequences of Haar null sets have Haar null limits.