Cfrgtkky

I was looking at the proof of the theorem stating that for two independent rv's $X,Y$ with density functions $f,g$ the density function of the rv $X+Y$ is the convolution of $f$ and $g$.

After having defined $A_z={(x,y) in mathbb R^2 : x+y leq z}$ for $z in mathbb R$ we get

$P(X+Y leq z) = P((X,Y) in A_z) = int_{A_z}f(x)g(y),dxdy=int_{-infty}^{infty}, dx ,f(x) int_{-infty}^{z-x}, dy , g(y), = , ...$

Can someone please explain the last equality
$int_{A_z}f(x)g(y),dxdy=int_{-infty}^{infty}, dx ,f(x) int_{-infty}^{z-x}, dy , g(y)$? I don't understand how the boundaries have been chosen and why we can just take the functions out of the integrals. Thank you.

For fixed $x$ the inequality $x+y leq z$ is same as $y leq z-x$ so $y$ ranges from $-infty$ to $z-x$. Once the inequality $x+y leq z$ has been taken care of, there is no further restriction on $x$ so $x$ ranges from $-infty$ to $infty$. (You can also do it the other way around: keep $y$ fixed, integrate w.r.t. $x$ from $-infty$ to $z-y$ and then integrate w.r.t $y$).