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

Question:

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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