The proof goes something like this:
we take the integral of $f(t)*g(t)$, we do integration by parts, and then we get: $$ g(x)F(x)-g(a)F(a)-(\text{integral}).$$
Now $g(x)F(x)$ is $0$ because the limit of $g$ at infinity is $0$ and $F$ is bounded. But why is $g(a)*F(a)$ also $0$? I couldn't find an explanation for that online.
I was told it's because $F(a)$ is really the integral from $a$ to $a$ which is obviously zero, but it's not, is it? that would be $F(a)-F(a)$ by Newton-Leibniz. $F(a)$ is simply the value of the anti-derivative $F$ at $a$, which doesn't have to be zero at all.
Thanks in advance!