Say you have the double integral $\displaystyle \displaystyle \int_{0}^{\infty} \int_{0}^{\infty} f(x,y) \ dx \ dy $ and you want to change the order of integration.

Normally this can be done if the integrand is always positive and/or if the double integral converges absolutely. But I don't think that those are the only conditions under which a change is permitted. How else could you justify changing the order of integration?

Say that the integrand of the above double integral is not always posisitve and the double integral does not converge absolutely. Could you do the following?

$\displaystyle \displaystyle \int_{0}^{\infty} \int_{0}^{\infty} f(x,y) \ dx \ dy = \lim_{R \to \infty} \int_{0}^{R} \int_{0}^{\infty} f(x,y) \ dx \ dy $ where R is a value such that the integrand is now always positive.

$\displaystyle \displaystyle = \lim_{R \to \infty} \int_{0}^{\infty} \int_{0}^{R} f(x,y) \ dy \ dx = \lim_{R \to \infty} \int_{0}^{\infty} f(x,R) \ dx $

And now, if it's possible, justify bringing the limit inside of the integral by use of the dominated convergence theorem.