Show that $f$ is a measurable function if $f$ is bounded

Question

Let $m(E)=0$. Show that if $f$ is a bounded function on E, then $f$ is measurable, and the Lebesgue integral is zero. I was thinking of using this theorem that state that $\int f =0 $ iff $f=0$ that would means that $f$ is the zero function, which i don't think i can assume. any idea thanks