Interchanging a limit and an integral: necessary and sufficient conditions

Journal of inequalities and applications

article

Takashi Kamihigashi¹ 

[1] Center for Computational Social Science, Kobe University

关键词: σ -finite measure space;    Integrable functions;    Vitali convergence theorem;    Uniform integrability;   

DOI  :  10.1186/s13660-020-02502-w

学科分类：电力

来源: SpringerOpen

【 摘 要 】

Let $\{f_{n}\}_{n \in \mathbb {N}}$ be a sequence of integrable functions on a σ-finite measure space $(\Omega, \mathscr {F}, \mu )$ . Suppose that the pointwise limit $\lim_{n \uparrow \infty } f_{n}$ exists μ-a.e. and is integrable. In this setting we provide necessary and sufficient conditions for the following equality to hold: $$ \lim_{n \uparrow \infty } \int f_{n} \, d\mu = \int \lim_{n \uparrow \infty } f_{n} \, d\mu. $$.

【 授权许可】

CC BY