Product measures and Tonelli–Fubini

Take sigma-finite measure spaces (X,A,μ) and (Y,B,ν). Their product sigma algebra is generated by rectangles E×F with E in A and F in B. The product measure is characterized by assigning such a rectangle the product μ(E)ν(F), with the standard zero-times-infinity convention. Sigma-finiteness ensures the usual existence and uniqueness framework and the integration results used here. A set's sections, obtained by fixing one coordinate, are measurable when the set belongs to the product sigma algebra.

The product of two complete measure spaces need not itself be complete on the product sigma algebra. For Lebesgue integration on Euclidean product spaces, one often passes to the completion. Then section and iterated-integral statements may need almost-everywhere qualifications rather than claims about every exceptional section. In this lesson, the basic theorem is stated for product-measurable functions, and completed-space versions are interpreted through suitable representatives. Naming this convention prevents a technically false assertion that all sections of every completed measurable set are automatically measurable.

For a nonnegative product-measurable f, Tonelli identifies its product integral with either iterated integral, permitting the value infinity. The theorem is therefore useful as a test: apply it to |f| to decide whether a signed function is integrable. If that integral is finite, Fubini applies to f itself. Almost every section is then integrable, the section integral defines an integrable function of the other coordinate, and either integration order returns the same finite answer. Exceptional sections do not alter the product integral.

This logic also governs infinite sums. Counting measure on the natural numbers is sigma-finite, so summing a nonnegative array in either order is Tonelli on a discrete product space. For a signed array, absolute summability gives Fubini and permits reordering. Merely obtaining finite answers in each iterated order is not enough to establish absolute integrability. It is safer to inspect |f| before simplifying a signed expression, because cancellation can hide exactly the divergence that invalidates the exchange.

On (0,1)² consider f(x,y)=(x²−y²)/(x²+y²)². For fixed positive x, integrating in y uses the derivative of y/(x²+y²) and gives 1/(1+x²). The resulting integral in x is pi/4. Reversing the roles uses the negative derivative of x/(x²+y²), yielding negative pi/4 instead. Both iterated answers exist, but they disagree. That disagreement cannot occur under Fubini's absolute-integrability hypothesis, so a singularity has defeated the proposed permission.

Polar coordinates explain the obstruction. Near the origin, the absolute value behaves as |cos(2θ)|/r² while area contributes r dr dθ. On an angular sector where the cosine stays bounded away from zero, the remaining radial integral is a positive multiple of the divergent integral of 1/r. Thus |f| is not integrable. A symmetric cutoff may suggest a principal value, but that is a different object with a specified limiting prescription, not the ordinary signed Lebesgue product integral.