This question seems like an easy measure theory problem but I don't know how to proceed.
Suppose $f$ is defined a.e., and suppose $g$ and $h$ are defined on all $\mathbb R^n$, and $g=f$ a.e. and $h=f$ a.e. Then prove that:
$g$ is measurable $\iff$ $h$ is measurable.
I think this is true if and only if notation is trivial but how to prove?