Uniform Fatou's lemma

Abstract

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.

Sufficiency of Deterministic Policies for Atomless Discounted and Uniformly Absorbing MDPs with Multiple Criteria

“…The function is negative because π(a|x) > 0 for all a ∈Ã(x), the difference in the second integral is nonpositive for all a ∈Ã(x), and this difference is negative for some a ∈Ã(x), where x ∈ X < \ X > . Equalities (17) are proved. The equality v π * = v σ * holds for every policy π * in the submodel with action setsÃ(·) if and only if v σ = v π for every stationary policy σ in this submodel.…”

“…Since σ(·|x) π(·|x) for all x ∈ X, Lemma 4.2 and formulae (16), (17) imply that q σ (X \ X = ) = 0. Let σ n,π be the policy that follows σ at times t = 0, 1, .…”

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

“…{µ n } n=1,2,... respectively. As explained in [16], inequality (4.1) is stronger than the inequality in Fatou's lemma, and the sufficient condition in Proposition 4.1 can be viewed as the uniform version of Fatou's lemma. Since the convergence in (4.2) is a uniform version of convergence of integrals, the sufficient condition in Proposition 4.2 can be viewed as the uniform version of Lebesgue's dominated convergence theorem.…”

“…However, their versions for finite measures are also important for probability theory and its applications. This is the reason this paper and [16] study finite measures rather than probabilities. For example, the theory of optimization of Markov decision processes with multiple criteria is based on considering occupancy (also often called occupation) measures, which typically are not probability measures [4].…”

“…This is important for applications. For this reason we provide in Section 4 alternative formulations of the uniform Fatou lemma and Lebesgue convergence theorem from [16] and two classic facts important for applications. In these formulations uniform integrability is substituted with asymptotic uniform integrability.…”