Dear readers,
I want to write down a rigorous proof of the intuitive fact:
$\displaystyle \int^{\infty}_{0}\int^{x}_{0} \frac{dF(t)}{dt}dtdx =\int^{\infty}_{0}xF(x)dx$
Anyone knows how to formulate such a proof?
Thnx
No, there is no such proof because the "intuitive fact" you give is not true.
By the "Fundamental theorem of Calculus" $\displaystyle \int_0^x \frac{dF(t)}{dt}dtdx= F(x)- F(0)$.
$\displaystyle \int_0^\infty\int_0^x \frac{dF}{dt}dtdx= \int_0^\infty (F(x)- F(0))dx$
That does not exist unless F(0)= 0 in which case
$\displaystyle \int_0^\infty\int_0^x \frac{dF}{dt}dtdx= \int_0^\infty F(x) dx$