Can we use the Second Mean Value Theorem over infinite intervals? 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.

No. Consider $g(t)=\frac{1}{t}$, $f=g$ on $[1,\infty)$.
