Uniform Fatou's lemma

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:444

Article

Feinberg, Eugene A.¹  Kasyanov, Pavlo O.²  Zgurovsky, Michael Z.³ 

[1] SUNY Stony Brook, Dept Appl Math & Stat, Stony Brook, NY 11794 USA

[2] Natl Tech Univ Ukraine, Kyiv Polytech Inst, Inst Appl Syst Anal, Peremogy Ave 37,Build 35, UA-03056 Kiev, Ukraine

[3] Natl Tech Univ Ukraine, Kyiv Polytech Inst, Peremogy Ave 37,Build 1, UA-03056 Kiev, Ukraine

关键词: Finite measure;    Convergence;    Fatou's lemma;    Radon-Nikodym derivative;   

DOI  :  10.1016/j.jmaa.2016.06.044

来源: Elsevier

【 摘 要 】

Fatou's lemma is a classic fact in real analysis stating 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 lemma, and it holds under the minor assumption that all the integrals in the inequality are well-defined. The uniform Fatou lemma improves the classic Fatou lemma in the following directions: the uniform Fatou 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 lemma are formulated. The examples in this paper demonstrate that: (a) the uniform Fatou lemma may indeed provide a more accurate inequality than the classic Fatou lemma; (b) the uniform Fatou lemma does not hold if convergence of measures in total variation is relaxed to setwise convergence. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free