measure theory - $lim_{ntoinfty}int f_n dmu=0$ implies that $f_n to 0$ a.e $mu$

Let $(X,F,\mu)$ be finite measure space.

Let $f_n$ be sequence of measurable function from $X$ and $f_n\geq 0$ almost everywhere $\mu$.

Claim.

If $\lim_{n\to 0}$$\int_X f_n d\mu=0$, then $f_n$ converges to $0$ a.e $\mu$?

Intutively It's true.

Can you help me?