Relation between integration and measure

Let $(X,M,\mu)$ be (positive) measure space with $\mu(X)<\infty$. Let $f:X\to\mathbb{R}$ be measurable with $f(x)>0$ almost everywhere. Let $\{E_n\}\subseteq M$ with $\lim_n\int_{E_n}fd\mu=0$. Prove $\lim_n\mu(E_n)=0$.

My original thinking is since $f>0$ a.e., we may assume $f>0$ everywhere and find a positive lower bound for $f$, but unfortunately this is not true.