interchange summation and iterated integration

Question

A math article makes use of the following equality, without proof. $$ \int_{0}^{1}\int_{0}^{1} \sum_{n\ge 0} (xy)^n dx\, dy = \sum_{n\ge 0} \int_{0}^{1}\int_{0}^{1}x^n y^n dx\, dy$$

But $s_k(x,y) = \sum_{n=0}^{k} (xy)^n$ does not converges uniformly to $\frac{1}{1-xy}$, so we cannot leverage uniform convergence to interchange integral and summation.

Maybe some result of analysis that I don't know justifies the equality above. Could anyone give a pointer?

Uniform convergence is a strong condition for this kind of swap. You want to look at Fubini-Tonelli 's theorem. http://en.wikipedia.org/wiki/Fubini%27s_theorem — Evan, Aug 24 '13 at 05:19
http://en.wikipedia.org/wiki/Fubini's_theorem#Tonelli.27s_theorem — Cameron Williams, Aug 24 '13 at 05:20

score 1 · Accepted Answer · answered Aug 24 '13 at 05:24

1

Such a change of order is always justified when the function being summed / integrated is nonnegative, and the space is sufficiently "nice," by Fubini's Theorem.

answered Aug 24 '13 at 05:24

interchange summation and iterated integration

1 Answers1