Let $(X,F,u)$ be a measure space and $f,g$ measurable functions. Show that if $f$ is integrable and $g$ is bounded and measurable, then $fg$ is integrable.
$f$ is integrable (on a set $E$) means that $\int_E|f|d\mu<\infty$. And we want to show that $\int_E|fg|d\mu<\infty$. But if $|g(x)|<M$ for all $x$, then the function $|fg|$ is bounded from above by $M|f|$, so we have $$\int_E|fg|d\mu\leq\int_EM|f|=M\int_E|f|<\infty.$$
So we don't need the condition that $g$ is measurable, do we?
We do need that condition. The Lebesgue integral simply isn't defined for nonmeasurable functions. If $g$ is not measurable, it might not make sense to take the integral of $|fg|$.