The following observation demonstrates the necessity of the $\sigma$ -finite assumption in Tonelli's and Fubini's theorem. Let $X$ denote the closed unit interval $[0,1]$ equipped with Lebesgue measure and $Y$ the same set, but this time equipped with counting measure $\nu$ . Let

Also observe that there does not exist a simple function on $X\times Y$ that is dominated by $f$ . Hence,$$ \int_{X\times Y} f(x,y) d (\mu(x)\times \nu(y) = 0$$ Therefore, the integrand is $L^1$ integrable relative to the product measure. However, as we observed above, the iterated integrals do not agree. This observation illustrates the need for the $\sigma$ -finite assumption for Fubini's theorem.