Past exam of the integration course of the University of Oxford / 2014 / 3 / a

Let $(X,\mathcal{A},\mu )$ and $(Y,\mathcal{B},\nu )$ be measure spaces with sigma-finite measures, and use their product measure.

Fubini's theorem: if $u$ is a measurable absolutely integrable function on $X\times Y$, then its sections are integrable for almost every fixed value of the other coordinate, the resulting section-integral functions are integrable, andAll three integrals are finite. Values of an inner integral on an exceptional null set can be assigned arbitrarily.

Tonelli's theorem: if $u:X\times Y\to [0,\infty ]$ is measurable and nonnegative, the same equality holds for its nonnegative product and iterated integrals, allowing the common value $+\infty$. No advance absolute integrability assumption is needed. On a completed Lebesgue measure product, the section and measurability assertions are interpreted almost everywhere.

Thus Tonelli's theorem can first establish that $\int |u|<\infty$; after this, Fubini's theorem permits signed integration in either order.