[THEOREMS] Fubini and iterated integrals

Fubini's theorem on a rectangle

If $f$ is continuous on

R=[a,b]\times[c,d],

then

\begin{aligned} \iint_R f(x,y)\,dA &=\int_a^b\left(\int_c^d f(x,y)\,dy\right)dx\\ &=\int_c^d\left(\int_a^b f(x,y)\,dx\right)dy. \end{aligned}

The inner integral computes one cross-section and the outer integral adds the cross-sections. In

\int_a^b\int_c^d f(x,y)\,dy\,dx,

the $dy$ says that $y$ is integrated first and the $dx$ says that $x$ is integrated second. Limits must belong to the variable whose differential they accompany.

The iterated formula remains valid for a bounded piecewise-continuous $f$ whose discontinuities lie on finitely many piecewise-smooth curves. This includes zero extensions whose only discontinuities lie on the region boundary.

Unbounded or improper integrals need stronger safeguards. If $f$ is measurable and $\iint_R|f|\,dA<\infty$ , Fubini's theorem gives the two iterated integrals and permits their order to be exchanged. If $f\ge0$ is measurable, Tonelli's theorem gives the same equality with values in $[0,\infty]$ , even when the integral is infinite. Similarly, if nonnegative functions increase pointwise to a limit, monotone convergence justifies passing the limit through the integral, as in the Basel sum.

An improper calculation should therefore first use bounded truncations (or a nonnegative convergence theorem). It must not subtract two separately divergent integrals.

Theorems from Fubini and iterated integrals

Fubini's theorem on a rectangle

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