Finiteness In Tonelli's And Fubini's Theorems Why do we suppose in Tonelli's Theorem that  $(X,Σ_X , μ_x)$ and  $(Y,Σ_Y , μ_y)$ to be σ-finite? Does the theorem fail when it is not? If yes, could someone provide an example?
We also suppose in Fubini's theorem that : 
$\int_{X×Y}|f(x,y)|$ $d(μ_x×μ_y) < \infty$
Does that mean it is not true when the integral = $\infty$ ? If yes, could someone provide an exmaple?

My favorite again involves the product of Lebesgue measure and counting measure, both on $[0,1]$. Let $f$ be the characteristic function of the diagonal. Then one of the iterated integrals is $0$, the other is $1$, while the integral with respect to the product measure is $\infty$.
