# Math Help - Fubini's theorem-violation of the integrability condition

1. ## Fubini's theorem-violation of the integrability condition

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

Why is

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

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

Can someone show me how to compute them?

2. ## Re: Fubini's theorem-violation of the integrability condition

I have figured that out