A property of a kind of product integral

Question

Please, could you say to me which property was utilized here?

$$\int_0^\infty \int_0^x f(a)g(x-a)\,da\, dx=\int_0^\infty f(a)\,da\int_0^\infty g(x)\, dx$$

Many thanks!

Answer 1

$\int_0^\infty \int_0^x f(a)g(x-a)\,da\, dx$=$\int_0^\infty \int_a^\infty f(a)g(x-a)\,dx\, da=\int_0^\infty f(a)\,da\int_0^\infty g(x)\, dx$

The first "=" is due to Fubini and $a\leq x$, you integrate both sides over the set $\{(a,x)\in \mathbb{R}^2 : 0\leq a\leq x\}$