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

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• 作者：Takashi Kamihigashi
• 关键词：Fatou’s lemma ; σ ; finite measure space ; infinite ; horizon optimization ; hyperbolic discounting ; existence of optimal paths
• 刊名：Journal of Inequalities and Applications
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1536KB
• 刊物主题：Analysis; Applications of Mathematics; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1029-242X
• 卷排序：2017

文摘

Given a sequence \(\{f_{n}\}_{n \in \mathbb {N}}\) of measurable functions on a σ-finite measure space such that the integral of each \(f_{n}\) as well as that of \(\limsup_{n \uparrow\infty} f_{n}\) exists in \(\overline{\mathbb {R}}\), we provide a sufficient condition for the following inequality to hold: $$ \limsup_{n \uparrow\infty} \int f_{n} \,d\mu\leq \int\limsup_{n \uparrow\infty} f_{n} \,d\mu. $$ 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.KeywordsFatou’s lemmaσ-finite measure spaceinfinite-horizon optimizationhyperbolic discountingexistence of optimal paths1 IntroductionLet \((\Omega, \mathscr {F}, \mu)\) be a measure space. Let \(\mathcal {L}(\Omega)\) be the set of measurable functions \(f: \Omega\rightarrow\overline{\mathbb {R}}\). A standard version of (reverse) Fatou’s lemma states that given a sequence \(\{f_{n}\}_{n \in \mathbb {N}}\) in \(\mathcal {L}(\Omega)\), if there exists an integrable function \(f \in \mathcal {L}(\Omega)\) such that \(f_{n} \leq f\)μ-a.e. for all \(n \in \mathbb {N}\), then $$\begin{aligned} \mathop {\overline {\lim }}_{n \uparrow\infty} \int f_{n} \,d\mu\leq \int \mathop {\overline {\lim }}_{n \uparrow\infty} f_{n} \,d\mu, \end{aligned}$$ (1.1) where \(\mathop {\overline {\lim }}= \limsup\). We call the above inequality the Fatou inequality.Some sufficient conditions for this inequality weaker than the one described above are known. In particular, provided that the integral of each \(f_{n}\) as well as that of \(\mathop {\overline {\lim }}_{n \uparrow\infty} f_{n}\) exists, ‘uniform integrability’ of \(\{f_{n}^{+}\}\) (where \(f_{n}^{+}\) is the positive part of \(f_{n}\)) is a sufficient condition for the Fatou inequality (1.1) in the case of a finite measure (e.g., [1–4]); so is ‘equi-integrability’ of the same sequence in the case of a σ-finite measure (see [5, 6]). These conditions are precisely defined in Section 2.