A generalization of Fatou’s lemma for extended real-valued functions on σ-finite measure spaces: with an application to infinite-horizon optimization in discrete time

Abstract

Given a sequence of measurable functions on a σ-finite measure space such that the integral of each as well as that of exists in , we provide a sufficient condition for the following inequality to hold: Our condition is considerably weaker than sufficient conditions known in the literature such as uniform integrability (in the case of a finite measure) and equi-integrability. As an application, we obtain a new result on the existence of an optimal path for deterministic infinite-horizon optimization problems in discrete time.

Interchanging a limit and an integral: necessary and sufficient conditions

“…) is uniform convergence on each A i ; see (2.5) below. Condition (2.4) is somewhat similar to some of the conditions (such as uniform integrability) used in the known results reviewed in Sect 4…”

“…As in [4], we say that a sequence {A i } i∈N in F is a σ -finite exhausting sequence if it is exhausting and…”

“…Hence it follows by[4, Lemma 8.4] that {A i } is a σ -finite exhausting sequence. Finally, the second conclusion holds since (2.5) implies (2.3).…”