Suppose $(X,S,\mu)$ is a measure space, $f$ is a non-negative measurable function, $E\in S$, and $\mu(E)=0$.
I want to show that $\int_E f d\mu=0$.
$\int_E f d\mu:=\int_X f\chi_E d\mu = sup\{\int_Xu\chi_E d\mu\mid 0\leq u\leq f$ and $u$ is a simple function $\}$.
It suffices to show that $\int_X u\chi_Ed\mu=0$ for each simple function $u$, which seems obvious but I'm not sure how to show it.