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, $\int\int f d\mu d\nu$ and $\int\int f d\nu d\mu$ exist and are unequal.
It seems to me that $\int\int f d\mu d\nu = \sum_n\sum_m f(m,n) = \sum_n f(n,n) + f(n+1,n) = \sum_n 1-1 = \sum_n 0 = 0$ and $\int\int f d\nu d\mu = \sum_m\sum_n f(m,n) = \sum_m f(m,m) + f(m,m-1) = \sum_m 1-1 = \sum_m 0 =0$. So the two integrals are equal.
What am I doing wrong? Thanks!