Let $\displaystyle f$ be nonnegative measurable function. Show $\displaystyle \int f = 0$ implies $\displaystyle f=0$ almost everywhere.

let $\displaystyle E=\{x:f>0\}$. Then $\displaystyle E=\bigcup E_n$ where $\displaystyle E_n=\{x:f>1/n \}$.

now 0=$\displaystyle \int f \geq \int f1_{E_n} \geq \frac{1}{n}\int 1_{E_n} = \frac{1}{n} mE_n$.

for each $\displaystyle n$, $\displaystyle 1/n \neq 0$ so $\displaystyle mE_n=0$.

Here is my question. i am not sure if i can say $\displaystyle lim_{n \rightarrow \infty} mE_n = 0$. can i say that?