Uniform Fatou's Lemma
Uniform Fatou's Lemma
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统一 Fatou 引理
DOI：
10.1016/j.jmaa.2016.06.044
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
M. Zgurovsky
中科院分区：
文献类型：
--
作者：
E. Feinberg;P. Kasyanov;M. Zgurovsky
Fatou引理是真实的分析中的一个经典事实，它指出函数积分的下极限大于或等于下极限的积分。本文引入了一个更强的不等式，该不等式对可测空间的可测子集上的积分一致成立。给出了有限测度序列依全变差收敛时该不等式成立的充要条件。这个陈述被称为一致法图引理，并且在小的假设下成立，即不等式中的所有积分都是定义良好的。一致Fatou引理在以下几个方面改进了经典的Fatou引理：一致Fatou引理陈述了一个更精确的不等式，它提供了一个充要条件，它处理了可变测度.各种推论的一致Fatou引理制定。本文的例子表明：（a）一致Fatou引理确实可以提供比经典的Fatou引理更精确的不等式;（B）当测度按全变差收敛放宽到按集收敛时，一致Fatou引理不成立.
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.