Fubini-Tonnelli's theorem

Warning

The $\sigma$-finiteness is crucial, consider the simple counterexaple with measure space $([0,1],\mathcal{B},\lambda )$ and $([0,1],\mathcal{B},\#)$ (the first is the usual lebesgue measure, the second is the counting measure). If we try to integrate the characteristic function of the diagonal set:

$$D:=\{(x,x):x\in [0,1]\}$$ $${\int }_{[0,1]}{\int }_{[0,1]}{1}_D(x,y)d\lambda (x)d\#(y)=0$$

on the other hand

$${\int }_{[0,1]}{\int }_{[0,1]}{1}_D(x,y)d\#(y)d\lambda (x)=1$$