A generalization of Fatou's lemma for extended real-valued functions on sigma-finite measure spaces: with an application to infinite-horizon optimization in discrete time

作者：Kamihigashi Takashi^*

来源：JOURNAL OF INEQUALITIES AND APPLICATIONS, 2017, 2017(1): 24.

DOI：10.1186/s13660-016-1288-5

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

Given a sequence {f(n)}(n epsilon N) of measurable functions on a sigma-finite measure space such that the integral of each fn as well as that of lim sup(n up arrow infinity) f(n) exists in (R) over bar, we provide a sufficient condition for the following inequality to hold:
lim sup (n up arrow infinity) integral f(n) d mu <= integral lim sup (n up arrow infinity) 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.

• 出版日期2017-1-18