I am studying the proof of the following theorem in Rudin's book (RCA).
at the beginning of the proff it is necessary to prove that the function
$\hspace{3cm} F(x,y) := f(x-y)g(y)$
is a Borel function on ${\mathbb R}^2$ for to be able to use Fubini's theorem.
my doubt is this:
why F being a Borelian function implies that we can use Fubini's theorem?
by hypothesis of Fubini's theorem, it is only necessary to show that
$\hspace{3cm} F $ is $ (m\times m) $- measurable, where $m$ is the Lebesgue measure.
Is that what I should prove? how does this follow from the fact that F is Borelian?