 Fubini's theorem

Let $(X,\mathcal{A},\mu )$ and $(Y,\mathcal{B},\nu )$ be sigma-finite measure spaces, and let $f$ be measurable for the product sigma-algebra. Ifthen its sections are integrable for almost every fixed coordinate, the resulting section integrals are integrable, andValues on the exceptional null sets can be assigned arbitrarily. The complex-valued version follows by real and imaginary parts.^[ref]

A useful order of reasoning is to apply Tonelli theorem to $|f|$ first. Once that nonnegative double integral is finite, Fubini justifies either signed order. For $f(x,y)=xy$ on $[0,1{]}^2$, absolute integrability is immediate and either order gives $1/4$. Merely obtaining two finite iterated signed integrals does not prove joint integrability; finite iterated integrals without joint integrability gives an explicit counterexample. The absolute-value hypothesis prevents hidden cancellation of infinite positive and negative contributions.