Fatou's Lemma for Weakly Converging Probabilities

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.

Uniform Fatou's lemma

“…, inequality (2.1) is the uniform version of inequality (2.6) of Fatou's lemma. There are generalized versions of Fatou's lemma for weakly and setwise converging sequences of measures; see Royden [18, p. 231], Serfozo [21], Feinberg et al [10], and the references in [10]. In particular, according to Royden [18, p. 231], for measures μ (n) converging to μ setwise, Fatou's lemma has the same formulation as the classic Fatou lemma with inequality (2.6) modified…”

“…In particular, Fatou's lemma holds for setwise and weakly converging measures. For setwise converging measures it is presented in Royden [18, p. 231] for nonnegative functions and in Feinberg et al [10] for functions that can take negative values. For weak convergence, it was introduced by Serfozo [21] for locally compact spaces.…”

“…As was observed by Schäl [19], the local compactness assumption is not needed. Feinberg et al [10] established Fatou's lemma for weakly converging measures and possibly negative functions. In the case of weak convergence, the lower limit of functions should be defined in a stronger sense than in the setwise convergence case and in the classic case of a single measure.…”

“…], Serfozo[21], Feinberg et al[10], and references therein.Example 4.2 demonstrates that, if convergence in total variation of finite measures {μ (n) } n=1,2,... to μ in Corollary 2.9 is relaxed to setwise convergence, equality (2.11) implies neither statement (i) nor statement (ii) from Theorem 2.1, and therefore neither statement (i) nor statement (ii) from Corollary 2.9 holds. Thus, inequality (2.1) does not yield either statement (i) or statement (ii) from Theorem 2.1, if the convergence in total variation of finite measures {μ (n) } n=1,2,... to μ in Theorem 2.1 is relaxed to setwise convergence.…”

Partially Observable Total-Cost Markov Decision Processes with Weakly Continuous Transition Probabilities

“…. ., such that z (n) → z weakly and a (n) → a, 16) that is, the family of functions R O 1 \O 2 is equicontinuous at all the points (z, a) ∈ P(X) × A.…”