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 {fn}n∈ℕ of measurable functions on a σ-finite measure space such that the integral of each fn as well as that of lim supn↑∞fn exists in , we provide a sufficient condition for the following inequality to hold: lim supn↑∞∫fndμ≤∫lim supn↑∞fndμ. 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.