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

Abstract 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 $\limsup_{n \uparrow\infty} f_{n}$ 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 . $$ \limsup_{n \uparrow\infty} \int f_{n} \,d\mu\leq \int\limsup_{n \uparrow\infty} f_{n} \,d\mu. $$ 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.

Keywords

• Fatou’s lemma
• σ-finite measure space
• infinite-horizon optimization
• hyperbolic discounting
• existence of optimal paths