In using Fubini's theorem, I always have problem on how change the variables when I switch the integrals. One example is the integral below. Suppose that all Fubini's conditions are satisfied and $Q(0) = 0$ and $Q'$ (derivative of $Q$) is finite. My issue is how I change the variable $x$ when I switch the integrals:
$\int_{0}^{\infty}(\int_{x}^{\infty} dp)Q'(x)dx = ???$
And by $dp$ I mean the probability measure. I want to do this:
$\int_{x}^{\infty}(\int_{0}^{\infty} Q'(x)dx)dp = ???$ but I don't know how to change the range of the first integral from $x$ to $\infty$. I think the correct one is from $0$ to $x$ but I don't know how to justify it.
Thanks for your help.
