Show that $\int f=0$ implies $f=0$ a.e, given that $f$ is a nonnegative measurable function. I am trying to show that $\int f=0$ implies $f=0$ a.e, given that $f$ is a nonnegative measurable function.
But I search my head from what I have learn I have no any clue of solving the problem. That is why I brought it to this room with the hope that somebody will give me a hint. 
Thanks 

Hint: $\{x: f(x)>0\}=\bigcup_n \{x: f(x)\geq \frac 1n\}$. So it suffices to show that $\{x: f(x)\geq\frac 1n\}$ has measure $0$.
