Fatou's Lemma for Weakly Converging Probabilities

TL;DR

This paper extends Fatou's lemma to scenarios involving sequences of measures that converge weakly, providing new inequalities for integrals with respect to varying measures.

Contribution

It introduces a version of Fatou's lemma applicable to sequences of measures that converge weakly, broadening its applicability.

Findings

Established inequalities for integrals under weakly converging measures

Extended classical Fatou's lemma to a new measure convergence context

Provides theoretical foundation for analysis involving weak measure convergence

Abstract

Fatou's lemma states under appropriate conditions that the integral of the lower limit of a sequence of functions is not greater than the lower limit of the integrals. This note describes similar inequalities when, instead of a single measure, the functions are integrated with respect to different measures that form a weakly convergent sequence.