Convergence of $\int_{A_n} f$ to $0$

Question

I am looking for a name or a reference in a textbook for the following result in order to quote it.

For any $f\in L^1(\mathbb{R})$-integrable function, we have $$\lim_{n\to\infty}\int_{A_n} f(x)dx=0,$$ whenever $(A_n:n\geq0)$ is a sequence of measurable set with $\mathcal{L}^1(A_n)\to0$.

It would be clear if we had $1_{A_n}\to0$ a.s. but it not necessarily the case.

Answer 1

It's a consequence of the dominated convergence theorem. The sequence of functions $\{f 1_{A_n}\}$ converges to 0 in measure and is dominated by the integrable function $|f|$.

Answer 2

The convergence in measure implies the convergence a.e of a subsequence.

Consequently, the 'classical' dominated convergence theorem can be extended to the case where only a convergence in measure is provided.