Question about Fatou's lemma

Question

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 $E\in \mathcal{M}$,

$$\int_E f~ d\mu \leq \liminf \int_E f_n ~d\mu\leq \liminf \int_X f_n ~d\mu = \int_X f ~d\mu$$

Answer 1

Hint: take an example where $f_n \to f$ a.e. on some $(X_1, {\cal M}_1, \mu_1)$ but $\int_{X_1} f_n \ d\mu_1$ does not converge to $\int_{X_1} f \ d\mu_1$. On another measurable space $(X_2, {\cal M}_2, \mu_2$ take $f_n = f$ with $\int_{X_2} f\ d\mu_2 = \infty$.
Consider the disjoint union of the two measurable spaces.