A note on the weak convergence of continuously integrable sequences

TL;DR

This paper explores the weak convergence of uniformly continuously integrable sequences of functions, providing a new proof that connects integrability conditions with measure convergence theories.

Contribution

It introduces a novel proof linking continuous integrability to weak convergence, replacing traditional tightness conditions with integrability assumptions.

Findings

Relates continuous integrability to relative compactness in the weak topology

Provides a detailed proof connecting measure convergence and integrability

Replaces tightness conditions with continuous integrability in convergence proofs

Abstract

A uniformly continuously integrable sequence of real-valued measurable functions, defined on some probability space, is relatively compact in the $\sigma(L^1,L^\infty)$ topology. In this paper, we link such a result to weak convergence theory of bounded measures as exposed in Billingsley (1968) and in Lo(2021) to offer a detailed and new proof using the ideas beneath the proof of prohorov's theorem where the continuous integrability replaces the uniform or asymptotic tightness.