Proof that the limit of a particular sequence of Lebesgue integrals equals zero.

Question

Prove that if $f \in L^{1}(A)$ and {${A_{n}}$} is a sequence of measurable subsets of A with $\lim_{n \to \infty}m(A_{n})=0$ then $lim_{n \to \infty} \int_{A_n}f=0$

I know there exists a very similar post for tackling this particular problem but I cannot find the complete proof of it. I would be grateful if someone could help.

Answer 1

$\lvert\int f1_{A_n}\,dm\rvert\leq\int\lvert f1_{A_n}\rvert\,dm\leq m(A_n)\int\lvert f\rvert\,dm\to0$ by Holder's inequality.

Answer 2

It's easy. For $n \in{} \mathbb{N}$, $\int_{A_n} f=\int_{A} f \mathbb{1}_{A_n}$. By Holder's inequallity, we have $\left |{\int_{A} f \mathbb{1}_{A_n}}\right |^2 \leq \int_{A} f^2 \int_{A} \mathbb{1}_{A_n}^2= \int_{A} f^2 \int_{A} \mathbb{1}_{A_n}=m(A_n)\int_{A} f^2 \xrightarrow{ n \to \infty } 0$ since $\lim_{ n \to \infty } m(A_n)=0$ and $f \in L^1(A)$. We conclude $\lim_{ n \to \infty } \int_{A} f \mathbb{1}_{A_n}=0$.