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

Journal of Inequalities and Applications, Jan 2017

Given a sequence { f n } n ∈ N of measurable functions on a σ-finite measure space such that the integral of each f n as well as that of lim sup n ↑ ∞ f n exists in 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 . 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.

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Takashi Kamihigashi. 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, Journal of Inequalities and Applications, 2017, 24, DOI: 10.1186/s13660-016-1288-5