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

中文翻译：

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交换极限和积分：必要和充分的条件

令$ \ {f_ {n} \} _ {n \ in \ mathbb {N}} $是σ有限度量空间$（\ Omega，\ mathscr {F}，\ mu）$上的可积函数序列。假设逐点极限$ \ lim_ {n \ uparrow \ infty} f_ {n} $存在μ-ae并且是可积的。在此设置中，我们为满足以下相等条件提供了必要和充分的条件：$$ \ lim_ {n \ uparrow \ infty} \ int f_ {n} \，d \ mu = \ int \ lim_ {n \ uparrow \ infty} f_ {n} \，d \ mu。$$