On the validity of Tonelli's Theorem

This work analyzes the validity of Tonelli’s theorem, overcoming the traditional assumption of $\sigma$-finiteness. We prove that the existence and equality of iterated integrals for indicator functions is a necessary and sufficient condition to define a product measure that allow to extend Tonelli’s Theorem to more general measure spaces, including $s$-finite measures. Finally, we characterize the semi-finiteness of such a product measure.