Measure Theory , Lp spaces measurable sets

Question

If $f\in L_{p}, 1<p<\infty$ and $({A}_{n})$ is a sequence of sets in $\mathcal{M}$ such that $\mu({A}_{n})\rightarrow 0$ as $n\rightarrow \infty$. Show that $\int_{{A}_{n}}{}{fd\mu}\rightarrow0$ as $n\rightarrow \infty$.

Answer 1

\begin{align*} \int_{A_{n}}|f|\dfrac{d\mu}{\mu(A_{n})}\leq\left(\int_{A_{n}}|f|^{p}\dfrac{d\mu}{\mu(A_{n})}\right)^{1/p}, \end{align*} so \begin{align*} \int_{A_{n}}|f|d\mu\leq\mu(A_{n})^{1-1/p}\left(\int_{A_{n}}|f|^{p}d\mu\right)^{1/p}\leq\mu(A_{n})^{1-1/p}\|f\|_{L^{p}}\rightarrow 0. \end{align*}

Answer 2

Idea:¿?

If $({A}_{n})$ is a sequence of sets in $\mathcal{M}$ such that $\mu({A}_{n})\rightarrow 0$ as $n\rightarrow \infty$ then $\exists N \in \mathbb{N} $ such that $\forall n\geq N$, $\mu({A}_{n})<\delta,\forall \delta >0$

Also we have , I do not know how to connect the hypothesis that the function is in ${L}_{p}$, to conclude the request