Fatou's Lemma for Weakly Converging Measures under the Uniform Integrability Condition

TL;DR

This paper extends classical convergence theorems like Fatou's lemma and Lebesgue's dominated convergence to sequences of measures that converge weakly, under conditions of uniform integrability, providing new formulations and connections to Young measures.

Contribution

It introduces new formulations of uniform Fatou's lemma and Lebesgue convergence theorem based on the equivalence of uniform integrability and asymptotic uniform integrability.

Findings

Extended Fatou's lemma for weakly converging measures.

New formulations of uniform Fatou's lemma and Lebesgue convergence.

Connections established with Dunford-Pettis theorem and Young measures.

Abstract

This note describes Fatou's lemma and Lebesgue's dominated convergence theorem for a sequence of measures converging weakly to a finite measure and for a sequence of functions whose negative parts are uniformly integrable with respect to these measures. The note also provides new formulations of uniform Fatou's lemma, uniform Lebesgue convergence theorem, the Dunford-Pettis theorem, and the fundamental theorem for Young measures based on the equivalence of uniform integrability and the apparently weaker property of asymptotic uniform integrability for sequences of functions and finite measures.