Uniform Fatou's Lemma

TL;DR

This paper introduces a uniform version of Fatou's lemma that provides a more precise inequality for integrals over measurable subsets, with conditions for its validity under measure convergence.

Contribution

It establishes the uniform Fatou's lemma, offering a stronger inequality, necessary and sufficient conditions, and applicability to variable measures.

Findings

The uniform Fatou's lemma can give more accurate bounds than the classic version.

It holds under total variation convergence of measures.

It does not hold under setwise convergence of measures.

Abstract

Fatou's lemma is a classic fact in real analysis that states that the limit inferior of integrals of functions is greater than or equal to the integral of the inferior limit. This paper introduces a stronger inequality that holds uniformly for integrals on measurable subsets of a measurable space. The necessary and sufficient condition, under which this inequality holds for a sequence of finite measures converging in total variation, is provided. This statement is called the uniform Fatou's lemma, and it holds under the minor assumption that all the integrals are well-defined. The uniform Fatou's lemma improves the classic Fatou's lemma in the following directions: the uniform Fatou's lemma states a more precise inequality, it provides the necessary and sufficient condition, and it deals with variable measures. Various corollaries of the uniform Fatou's lemma are formulated. The examples in this paper demonstrate that: (a) the uniform Fatou's lemma may indeed provide a more accurate inequality than the classic Fatou's lemma; (b) the uniform Fatou's lemma does not hold if convergence of measures in total variation is relaxed to setwise convergence.