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

摘要

Given a sequence { f n } n ∈ N ${f_{n}}_{n in mathbb {N}}$ of measurable functions on a σ-finite measure space such that the integral of each f n $f_{n}$ as well as that of lim sup n ↑ ∞ f n $limsup_{n uparrowinfty} f_{n}$ exists in R ‾ $overline{mathbb {R}}$ , we provide a sufficient condition for the following inequality to hold: lim sup n ↑ ∞ ∫ f n d μ ≤ ∫ lim sup n ↑ ∞ f n d μ . $$ limsup_{n uparrowinfty} int f_{n} ,dmuleq intlimsup_{n uparrowinfty} f_{n} ,dmu. $$ 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.