Let $(X,\mathcal S,\mu)$ be a measure space and $E\in \mathcal S$. For every $n\in \mathbb N$, $f_n:X\rightarrow \overline{\mathbb R}$ is measurable with $f_1\leq f_2\leq \dots$. Then $\int_E \lim \limits_{n\to\infty} f_n d\mu=\lim\limits_{n\to\infty}\int_E f_nd\mu$
if:
a) $\int_E f_1d\mu$ exists
b) $f_1\in \mathscr L^1(\mu,E)$ (this means statement a) is not enough)
c) even under the assumption b) not necessarily
I think b) is correct but I am not shure. How can I prove this?
