Let $[a,b]$ be any closed interval and let $f,g$ be continuous on $[a,b]$ with $g(x)\geq 0$ for all $x\in[a,b]$. Then the Second Mean Value Theorem says that

$$\int\_a^bf(t)g(t)\text{d}t = f(c)\int\_a^b g(t)\text{d}t,$$

for some $c\in(a,b)$.

Does this theorem work on the interval $[0,\infty]$ ?

EDIT: Assuming the integrals involved converge.