Passing to General Nonnegative Functions

From Simple Functions to General Functions

The product-integral identity is first established for nonnegative simple functions: integrating such a function on a product space gives the same value as integrating its section integrals. To pass from simple functions to a general nonnegative measurable function, we approximate it from below by an increasing sequence of simple functions. The Monotone Convergence Theorem then transfers the identity to the limit. No finiteness of the resulting integrals is required; either side may be \(+\infty\).

Throughout, let \((X,\mathcal{M},\mu)\) and \((Y,\mathcal{N},\nu)\) be sigma-finite measure spaces, not necessarily finite or complete, and equip \(X\times Y\) with \(\mathcal{M}\otimes\mathcal{N}\) and the product measure \(\mu\times\nu\). Functions in the main result are measurable and take values in \([0,\infty]\). For \(f:X\times Y\to[0,\infty]\), write \(f_x(y)=f(x,y)\). Sections of a product-measurable function are measurable, and the earlier result on nonnegative section integrals ensures that \(x\mapsto\int_Y f_x\,d\nu\) is measurable.

Increasing Simple Approximation

The approximation must work even when \(f\) is unbounded or takes the value \(+\infty\). For each integer \(n\ge1\), round \(f\) down to the grid of mesh \(2^{-n}\), while truncating at height \(n\). Explicitly, define \(q_n:[0,\infty]\to[0,\infty)\) and \(\varphi_n:X\times Y\to[0,\infty)\) by

Here \(\min(+\infty,n)=n\). Each \(\varphi_n\) is a finite-valued simple function: its possible values are among \(0,2^{-n},2\cdot2^{-n},\ldots,n\). It is measurable because it is obtained by applying a Borel measurable function \(q_n\) to the measurable function \(f\).

Proof. First, \(0\le q_n(t)\le t\) for every \(t\in[0,\infty]\), so \(0\le\varphi_n\le f\). To check that \(q_n(t)\le q_{n+1}(t)\), consider two cases. If \(t\le n\), then \(\min(t,n)=\min(t,n+1)=t\). Since every grid point \(k2^{-n}\) is also a grid point at level \(n+1\), rounding \(t\) down to the finer grid cannot decrease its rounded value. If \(t>n\), then \(q_n(t)=n\). When \(n<t<n+1\), the finer-grid rounded value is at least \(n\); when \(t\ge n+1\), \(q_{n+1}(t)=n+1\). Thus \(q_n(t)\le q_{n+1}(t)\) in all cases.

For finite \(t\), once \(n>t\), the truncation no longer changes \(t\), and \(0\le t-q_n(t)<2^{-n}\). Hence \(q_n(t)\to t\). For \(t=+\infty\), \(q_n(t)=n\), so \(q_n(t)\to+\infty\). Applying these pointwise statements to \(t=f(x,y)\) proves \(\varphi_n(x,y)\uparrow f(x,y)\) for every \((x,y)\). \(\square\)

The Identity for General Nonnegative Functions

Proof. Use the approximation lemma to choose nonnegative simple functions \(\varphi_n\uparrow f\) pointwise everywhere. For each \(n\), define \(A_n(x)=\int_Y\varphi_n(x,y)\,d\nu(y)\). The simple-function identity ensures that \(A_n\) is measurable and that \(\int_{X\times Y}\varphi_n\,d(\mu\times\nu)=\int_X A_n\,d\mu\).

Fix \(x\in X\). The sequence \(y\mapsto\varphi_n(x,y)\) is nonnegative, measurable, and increasing to \(y\mapsto f(x,y)\) at every \(y\). The Monotone Convergence Theorem on \((Y,\mathcal{N},\nu)\) therefore gives \(\displaystyle A_n(x)\uparrow A(x):=\int_Y f(x,y)\,d\nu(y)\). This holds for every \(x\), including when the limit is infinite. The function \(A\) is measurable by the result on nonnegative section integrals. Apply the Monotone Convergence Theorem on \((X,\mathcal{M},\mu)\) to \(A_n\uparrow A\), and on \((X\times Y,\mathcal{M}\otimes\mathcal{N},\mu\times\nu)\) to \(\varphi_n\uparrow f\). Taking limits in the simple-function identity yields \[ \int_{X\times Y} f\,d(\mu\times\nu) =\lim_{n\to\infty}\int_{X\times Y}\varphi_n\,d(\mu\times\nu) =\lim_{n\to\infty}\int_X A_n\,d\mu =\int_X A\,d\mu. \] All three limits are justified by Monotone Convergence; no interchange of a limit and an integral is being made without that theorem. The reversed-order identity follows by applying the same argument with the roles of \(X\) and \(Y\) exchanged. \(\square\)

Worked Applications

Why Monotonicity Matters

The proof depends on increasing approximations: Monotone Convergence applies to both the approximating functions on the product space and their section integrals. Pointwise convergence alone is not enough. For a concrete failure, on \(\mathbb{R}^2\) with product Lebesgue measure define \(g_n(x,y)=n^2\chi_{(0,1/n)\times(0,1/n)}(x,y)\). At every fixed \((x,y)\), \(g_n(x,y)\to0\): if either coordinate is nonpositive, the point is never in the indicated open square; otherwise, eventually \(1/n\le x\) or \(1/n\le y\). But \(\displaystyle\int_{\mathbb{R}^2}g_n\,d(m\times m) =n^2\cdot\frac1n\cdot\frac1n=1\) for every \(n\), whereas the integral of the pointwise limit is \(0\). The sequence is not increasing: at \((0.4,0.2)\), \(g_2=4\), since both coordinates are strictly between \(0\) and \(1/2\), while \(g_3=0\), since \(0.4\) is not strictly less than \(1/3\). Thus the increasing-sequence hypothesis cannot be omitted.

A second common error is to infer that the iterated integral is defined as a measurable function of the outer variable merely because each section is measurable. Measurability of the section \(f_x\) for each fixed \(x\) and measurability of \(x\mapsto\int_Y f_x\,d\nu\) are different assertions. The latter is supplied here by the result on nonnegative section integrals; it is needed before the outer integral is formed. The theorem concerns nonnegative measurable functions and permits infinite values. It does not, by itself, give the corresponding identity for signed or complex-valued functions, where cancellation and integrability conditions must be addressed separately.