Let $\displaystyle X_{1}=X_{2}=\mathbb{N}$ and let $\displaystyle \mu_{1},\mu_{2}$ be counting measures. Let $\displaystyle A=\{(k,k):k\in\mathbb{N}\}$ let $\displaystyle B=\{(k,k+1):k\in\mathbb{N}\}$ so A is the diagonal, B is off the diagonal. Let $\displaystyle f=\chi_{A}-\chi_{B}$

Why is

$\displaystyle \displaystyle\int_{X_{2}}(\int_{X_{1}}fd\mu_{1})d\ mu_{2}))=1$

while

$\displaystyle \displaystyle\int_{X_{1}}(\int_{X_{2}}fd\mu_{2})d\ mu_{1}))=0$

Can someone show me how to compute them?