Let ((X, mathscr{A}, mu)) be a measure space and (left(u_{n}ight)_{n in mathbb{N}} subset mathcal{L}^{1}(mu)) be a uniformly

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Let \((X, \mathscr{A}, \mu)\) be a measure space and \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{L}^{1}(\mu)\) be a uniformly convergent sequence.

(i) If \(\mu(X)<\infty\), then \(\lim _{n} \int u_{n} d \mu=\int \lim _{n} u_{n} d \mu\).

(ii) Assume that \(u=\lim _{n} u_{n} \in \mathcal{L}^{1}(\mu)\) and \(\lim _{n} \int u_{n} d \mu\) exists. It is true or false that \(\lim _{n} \int u_{n} d \mu=\int u d \mu\) ?

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