Interchanging a limit and an integral: necessary and sufficient conditions

[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: $$ \lim_{n \uparrow \infty } \int f_{n} \, d\mu = \int \lim_{n \uparrow \infty } f_{n} \, d\mu. $$.

[学科分类] 电力

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