Monotone convergence theorem for sets?

Question

Let $M = \bigcup\limits_{n=1}^\infty M_n$, $M_n$ be a measurable sets on $\mathbb R$ and $f$ be a measurable function on $\mathbb R$. Then I want to know how the equality $$ \lim\limits_{N\to \infty}\int_{M\setminus \bigcup\limits_{n=1}^NM_n} |f(x)| \,dx = 0 $$ holds. It seems something like monotone convergence theorem, but I know the monotone convergence theorem is for the monotone "functions". How can I justify this?

Answer 1

The map $\nu:A\mapsto \int_A |f|$ is a measure.

Note that $A_N:=M\setminus \bigcup_{n=1}^NM_n$ is a decreasing sequence that converges to $\emptyset$.

Assuming $\int |f|<\infty$, we have $\lim_N\nu(A_N)=\nu(\emptyset)=0$.

Answer 2

Hint: use characteristic functions to write each integral as an integral on $M$.