Assume that $f_n \to f$ almost everywhere, with $f_n$ integrable for all $n$ and $g$ is an integrable function such that $\lvert f_n \rvert \le g$.
(A) Then $f$ is integrable and -
$$\int f\,\mathrm d\mu = \lim_n \int f_n\,\mathrm d\mu.$$
And in fact a stronger condition (B) is true -
$$\int \left\lvert f_n - f \right\rvert\,\mathrm d\mu \to 0$$ as $n \to \infty$.
I have two questions on this.
- How does (B) imply (A)?
- How is (B) stronger than (A)?