A generalization of Fatou's lemma for extended real-valued functions onσ-finite measure spaces: with an application to infinite-horizon optimization in discrete time.
Given a sequence [Formula: see text] of measurable functions on aσ-finite measure space such that the integral of each [Formula: see text] as well as that of [Formula: see text] exists in [Formula: see text], we provide a sufficient condition for the following inequality to hold: [Formula: see text] 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.
RIEB, Kobe University, Rokkodai, Nada, Kobe 657-8501 Japan.
Takashi
T
Kamihigashi
Fatou’s lemma; existence of optimal paths; hyperbolic discounting; infinite-horizon optimization; σ-finite measure space