Prove $\lim_{n \to \infty}\mu(A_n) = 0$

Question

Let $(X, \mathcal{M}, \mu)$ be a measure space with $\mu(X)<\infty$ and $f:X\to \overline{\mathbb{R}}$ be a measurable function and $f>0$ a.e on $X$. Let $A_n\in\mathcal{M} ,\, n=1,2,...$ and suppose that $$\lim_{n\to\infty}\int_{A_n}fd\mu=0$$. Prove $$\lim_{n \to \infty}\mu(A_n) = 0$$