I am working on some exercises in Folland's real analysis. In number 2.48, they ask you to prove the following question:
Let $X = Y = \mathbb N$, $M = N = P(\mathbb N)$ and $\mu = \nu$ be counting measure on $\mathbb N$. Define $f(m,n) = 1$ if $m=n$, $f(m,n) = -1$ if $m = n+1$, and $f(m,n) = 0$ otherwise. Then, $\iint f\ \mathsf d\mu \mathsf d\nu$ and $\iint f\ \mathsf d\nu\mathsf d\mu$ exist and are unequal.
It seems to me that \begin{align}\iint f\ \mathsf d\mu\mathsf d\nu &= \sum_n\sum_m f(m,n)\\ &= \sum_n f(n,n) + f(n+1,n)\\ &= \sum_n 1-1\\ &= \sum_n 0 = 0\end{align} and \begin{align}\iint f\ \mathsf d\nu\mathsf d\mu &= \sum_m\sum_n f(m,n)\\ &= \sum_m f(m,m) + f(m,m-1)\\ &= \sum_m 1-1\\ &= \sum_m 0 =0.\end{align} So the two integrals are equal.
What am I doing wrong? Thanks!