$f$ measurable and bounded over a set of finite measure is integrable

Question

I came across a theorem that states that if $f$ is a bounded function defined on a measurable set $E$ with finite measure, then $f$ is measurable if and only if $f$ is Lebesgue integrable.

I was wondering how can we show the forward implication, i.e if $f$ is measurable then it is integrable.