Strengthening of weak convergence for Radon measures in separable Banach spaces

TL;DR

This paper proves that for any weakly converging sequence of sigma-finite Borel measures in a separable Banach space, there exists a compact embedded subspace where the measures are concentrated and also weakly converge.

Contribution

It establishes a new result linking weak convergence of measures to their concentration in a compact embedded subspace within separable Banach spaces.

Findings

Existence of a compact embedded subspace for measure concentration

Weak convergence is preserved within this subspace

Applicable to sigma-finite Borel measures in separable Banach spaces

Abstract

We prove in this short report that for arbitrary weak converging sequence of sigma-finite Borelian measures in the separable Banach space there is a compact embedded separable subspace such that this measures not only are concentrated in this subspace but weak converge therein.