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

Abstract

This paper describes Fatou's lemma 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 paper also provides new formulations of uniform Fatou's lemma, uniform Lebesgue's 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.