Past exam of the options II course of the University of Oxford / 2009 / D / D4 / i / Solution

On a product of sigma-finite measure spaces, Tonelli's theorem says that a nonnegative measurable function has equal product and iterated integrals, allowing $+\infty$.

Fubini's theorem says that if a measurable function is absolutely integrable on the product, almost every section is integrable, the integrals of sections are integrable, and both iterated integrals equal the product integral. In particular, to establish absolute integrability one can first apply Tonelli's theorem to the absolute value and show its iterated integral is finite.