Iterated Integration

Reading an Integral One Variable at a Time

The product measure from the previous tutorial lets us integrate functions on \(X\times Y\). Iterated integration is a way to compute such an integral by first fixing one coordinate, integrating over the other, and then integrating the result over the fixed coordinate. The order matters for how the calculation is set up, even when the theorems of this course guarantee that either order gives the same answer.

Throughout, take \((X,\mathcal{F},\mu)\) and \((Y,\mathcal{G},\nu)\) to be sigma-finite measure spaces, and take functions on \(X\times Y\) to be measurable with respect to \(\mathcal{F}\otimes\mathcal{G}\). The sigma-finiteness hypothesis is essential here: it is among the hypotheses of the Tonelli and Fubini theorems established earlier in this course. We will use those theorems by name rather than re-prove them.

The sections here are sections of a function: for fixed \(x\), the function of \(y\) is \(y\mapsto f(x,y)\), and for fixed \(y\), the function of \(x\) is \(x\mapsto f(x,y)\). The earlier section results for product-measurable sets, together with Tonelli’s Theorem, ensure that the inner integral of a nonnegative measurable function is measurable as a function of the remaining coordinate. In particular, the iterated integrals are well-defined as extended nonnegative integrals.

For a signed function, care is needed. Fubini’s Theorem applies when \(f\) is integrable on the product space, meaning \(\int_{X\times Y}|f|\,d(\mu\otimes\nu)<\infty\). It then gives section integrability almost everywhere and equality of both iterated integrals with the product-space integral. If the exceptional set of outer coordinates where a section is not integrable is nonempty, the inner integral can be assigned any value there without changing the outer integral. For a nonnegative function, Tonelli’s Theorem instead permits infinite values and does not require the product-space integral to be finite.

A Product of Separate Functions

A useful first case is a function that separates into one factor depending only on \(x\) and another depending only on \(y\). The result below turns a two-variable integral into a product of one-variable integrals. Its hypotheses ensure that the factors and their integrals are finite; this avoids ambiguity about multiplying zero and infinity.

Proof. The function \((x,y)\mapsto u(x)v(y)\) is product-measurable. For each fixed \(x\), nonnegative homogeneity of the integral gives $$ \int_Yu(x)v(y)\,d\nu(y)=u(x)\int_Yv(y)\,d\nu(y). $$ Integrating this equality over \(X\), and using nonnegative homogeneity once more, yields $$ \int_X\left(\int_Yu(x)v(y)\,d\nu(y)\right)d\mu(x) =\left(\int_Yv\,d\nu\right)\left(\int_Xu\,d\mu\right). $$ Tonelli’s Theorem applies because the spaces are sigma-finite and the product function is nonnegative and measurable. It identifies this iterated integral with the product-space integral and also identifies the iterated integral in the other order with that same value. This proves the result.

Changing the Order by Slicing a Region

For a nonrectangular region, the bounds in an iterated integral describe its sections. Fixing \(x\) gives a vertical slice; fixing \(y\) gives a horizontal slice. The following result formalizes this procedure by extending a function on a region to be zero outside it. It is a direct application of Fubini’s Theorem, not a new version of that theorem.

Proof. Extend \(f\) to \(F:X\times Y\to\mathbb{R}\) by setting \(F(x,y)=f(x,y)\) on \(E\) and \(F(x,y)=0\) outside \(E\). This extension is measurable: for any Borel set \(C\subseteq\mathbb{R}\), its preimage under \(F\) is assembled from the measurable part of \(E\) on which \(f\) lies in \(C\), and, if \(0\in C\), the measurable set \(E^c\). Also, $$ \int_{X\times Y}|F|\,d(\mu\otimes\nu)=\int_E|f|\,d(\mu\otimes\nu)<\infty. $$ Fubini’s Theorem therefore applies to \(F\). For each \(x\), \(F(x,y)\) equals \(f(x,y)\) on \(E_x\) and zero outside \(E_x\), so its section integral is the integral of \(f(x,\cdot)\) over \(E_x\), wherever that section is integrable. The corresponding statement holds for each horizontal section \(E^y\). Finally, the product-space integral of \(F\) is the integral of \(f\) over \(E\). Substituting these three identifications into Fubini’s equalities proves the result.

When the integrand is nonnegative and its integral may be infinite, use Tonelli’s Theorem in the same extension-by-zero method. The sigma-finiteness hypotheses on both factors still apply. For geometric calculations, it is especially important to describe the region itself before writing bounds: the bounds must include exactly the points of the region, up to boundary sets that do not affect the integral in the examples below.

Choosing an Order and Checking the Hypotheses

For a region described by inequalities, the practical task is to express each slice accurately. A reliable process is:

Changing the order is a choice of description, not permission to ignore the theorem’s hypotheses. In particular, for a signed function, separate iterated integrals may fail to exist or may not be interchangeable if absolute integrability is absent. For nonnegative functions on sigma-finite spaces, Tonelli allows either order, but one or both answers may be infinite. Always check measurability and the relevant integrability or nonnegativity condition before treating the two orders as equal.