Can I get help in solving this question.
Let $(X,\mathcal{M}, \mu)$ be a measurable space with $\mu$ be a positive measure. Let $f,f_n \geq 0$ be measurable functions such that $f_n\to f$ a.e. and $\lim_{n\to \infty} \int_X f_n~ d\mu = \int_X f ~d\mu$. Then if $\int_X f ~d\mu \lt \infty$, we have for every $E\in \mathcal{M}$, $\lim_{n\to\infty} \int_E f_n ~d\mu = \int_E f ~d\mu$.
I will want to find an example where $\int_X fd\mu = \infty$ and the conclusion fails.
These are my thoughts:
Since and $f, f_n\geq 0$ and measurable and $f_n\to f$ a.e. by Fatou's lemma, for any $E\in \mathcal{M}$,
$$\int_E f~ d\mu \leq \liminf \int_E f_n ~d\mu\leq \liminf \int_X f_n ~d\mu = \int_X f ~d\mu$$