Product Measures and Fubini

Multiple integrals established that iterated Riemann integration over a box computes the same number as the double integral, provided the integrand is continuous. That hypothesis is far stronger than necessary, and the Riemann framework cannot express the general statement at all. Here we build the product measure $\mu \otimes \nu$ on $X\times Y$ and prove the Tonelli and Fubini theorems, which say precisely when

$${\int }_{X\times Y}fd(\mu \otimes \nu )={\int }_X\left({\int }_{Y}f(x,y)d\nu (y)\right)d\mu (x)={\int }_Y\left({\int }_{X}f(x,y)d\mu (x)\right)d\nu (y).$$

Infinite products get their own treatment in §5: they are what make an infinite sequence of independent trials a legitimate object, and they are the reason probability can speak of i.i.d. sequences at all.

References: Folland, Real Analysis, §2.5; Rudin, Real and Complex Analysis, ch. 8.

1. The Product $\sigma$-Algebra

Definition. For measurable spaces $(X,\mathcal{A})$, $(Y,\mathcal{B})$, the product $\sigma$-algebra is $\mathcal{A}\otimes \mathcal{B}=\sigma (\{A\times B:A\in \mathcal{A},>B\in \mathcal{B}\})$.

The generating family — measurable rectangles — is a $\pi$-system (the intersection of two rectangles is a rectangle), which is what makes the uniqueness machinery of measure-theory.md §4 applicable throughout this page.

Proposition (sections). For $E\in \mathcal{A}\otimes \mathcal{B}$ every section $E_x=\{y:(x,y)\in E\}$ is in $\mathcal{B}$, and every $E^y$ is in $\mathcal{A}$. Likewise every section of a measurable $f$ is measurable.

Proof. The family of $E$ with all sections measurable is a $\sigma$-algebra containing the rectangles. $\square$

Sections are needed even to state Fubini: the inner integral ${\int }_{Y}f(x,y)d\nu (y)$ must make sense for fixed $x$.

Caution: Borel vs. completion. $\mathcal{B}({\mathbb{R}}^m)\otimes >\mathcal{B}({\mathbb{R}}^n)=\mathcal{B}({\mathbb{R}}^{m+n})$, but the corresponding statement for Lebesgue $\sigma$-algebras is false: ${\mathcal{L}}^m\otimes {\mathcal{L}}^n⊊{\mathcal{L}}^{m+n}$. The product of complete measures need not be complete, and sections of a ${\lambda }^{m+n}$-measurable set need not be measurable — only almost every section is. This is why the theorems below are stated for $\sigma$-finite measures on the product $\sigma$-algebra, with an "a.e. section" caveat in the completed case.

2. The Product Measure

Theorem. Let $\mu ,\nu$ be $\sigma$-finite. There is a unique measure $\mu \otimes \nu$ on $\mathcal{A}\otimes \mathcal{B}$ with

$$(\mu \otimes \nu )(A\times B)=\mu (A)\nu (B),$$

and for every $E\in \mathcal{A}\otimes \mathcal{B}$,

$$(\mu \otimes \nu )(E)={\int }_X\nu (E_x)d\mu (x)={\int }_Y\mu (E^y)d\nu (y).$$

Proof sketch. Rectangles form a $\pi$-system generating the product $\sigma$-algebra, and finite disjoint unions of rectangles form an algebra on which $(A,B)\mapsto \mu (A)\nu (B)$ is a premeasure; Carathéodory extension (measure-theory.md §5) gives existence, $\sigma$-finiteness plus the uniqueness corollary gives uniqueness. The displayed section formula is proved first for rectangles (where it is trivial) and then propagated by the $\pi$–$\lambda$ theorem, using monotone convergence for the increasing-union step. $\square$

$\sigma$-finiteness is essential. With $X=Y=[0,1]$, $\mu ={\lambda }^1$ and $\nu$ = counting measure, the diagonal $\Delta$ has $\int \nu ({\Delta }_x)d\mu =1$ but $\int \mu ({\Delta }^y)d\nu =0$: the two iterated integrals of an indicator disagree.

Lebesgue measure factors. ${\lambda }^{m+n}={\lambda }^m\otimes {\lambda }^n$ on Borel sets (both agree on boxes, then apply uniqueness). So the multi-dimensional Lebesgue measure of measure-theory.md §6 could equally have been defined as an $n$-fold product of ${\lambda }^1$.

3. Tonelli and Fubini

Tonelli's Theorem. Let $\mu ,\nu$ be $\sigma$-finite and $f:X\times Y\to >[0,\infty ]$ measurable. Then $x\mapsto {\int }_{Y}f(x,y)d\nu (y)$ is measurable, and the double and both iterated integrals are equal (all possibly $+\infty$).

Fubini's Theorem. Let $\mu ,\nu$ be $\sigma$-finite and $f\in L^1(\mu \otimes >\nu )$. Then $f(x,\cdot )\in L^1(\nu )$ for $\mu$-a.e. $x$, the a.e.-defined function $x\mapsto {\int }_{Y}f(x,y)d\nu (y)$ is in $L^1(\mu )$, and the double and both iterated integrals are equal.

Proof. Tonelli: true for indicators by §2, hence for simple functions by linearity, hence for all non-negative measurable $f$ by monotone convergence and simple approximation. Fubini: apply Tonelli to $f^{\pm }$ (or $|f|$), note both pieces are finite by hypothesis, and subtract; finiteness a.e. of the inner integral is exactly the statement that $\int (\int |f|d\nu )d\mu <\infty$. $\square$

The practical protocol. The two theorems are used as a pair, in this order:

1. Tonelli on $|f|$ — always permitted, no hypothesis to check — to decide whether $\iint |f|<\infty$.
2. If yes, Fubini on $f$ to exchange the order freely.

Skipping step 1 is the standard error. On $(0,1{)}^2$ the function $f(x,y)=\frac{x^2-y^2}{(x^2+y^2{)}^2}$ gives iterated integrals $\pi /4$ and $-\pi /4$; it is measurable and both iterated integrals exist, but $\iint |f|=>\infty$, so no theorem was ever available.

Two standard corollaries, both used constantly downstream:

• Layer-cake formula. For measurable $f\ge 0$ on $(X,\mathcal{A},\mu )$, $${\int }_{X}fd\mu ={\int }_0^{\infty }\mu (\{f>t\})dt,$$ by applying Tonelli to $1_{\{(x,t):t<f(x)\}}$ on $X\times (0,\infty )$. In probability this is the identity $E[X]={\int }_0^{\infty }P(X>t)dt$ for $X\ge 0$ (expectation.md).
• Convolution. For $f,g\in L^1({\mathbb{R}}^n)$, $(f\ast g)(x)=\int f(x-y)g(y)dy$ is defined for a.e. $x$, lies in $L^1$, and $\Vert f\ast g{\Vert }_1\le \Vert f{\Vert }_1\Vert g{\Vert }_1$ — Tonelli on $|f(x-y)g(y)|$ plus translation invariance. Convolution is the operation behind sums of independent random variables and behind mollification.

4. Change of Variables

Theorem. Let $U,V\subseteq {\mathbb{R}}^n$ be open and $T:U\to V$ a $C^1$ diffeomorphism. For measurable $f\ge 0$ (or $f\in L^1(V)$),

$${\int }_{V}f(y)dy={\int }_{U}f(T(x))|\det DT(x)|dx.$$

This is the Lebesgue-theoretic form of the substitution rule proved for Jordan-measurable regions in multiple-integrals.md §4. Two gains: no regularity assumption on the region, and the linear case ${\lambda }^n(TA)=|\det T|{\lambda }^n(A)$ is now a statement about all measurable sets. Combined with §2 it is the workhorse for polar/spherical coordinates and for Gaussian integrals.

5. Infinite Products

Finite products iterate without comment: $({\mu }_1\otimes {\mu }_2)\otimes {\mu }_3={\mu }_1\otimes {\mu }_2\otimes {\mu }_3$. Countable products need a hypothesis, and the natural one is probability.

Theorem (infinite product of probability measures). Let $(X_k,{\mathcal{A}}_k,>P_k{)}_{k\in \mathbb{N}}$ be probability spaces. On $X={\prod }_k{X}_k$ with the product $\sigma$-algebra ${⨂}_k{\mathcal{A}}_k$ — generated by the cylinder sets depending on finitely many coordinates — there is a unique probability measure $P={⨂}_k{P}_k$ with

$$P(A_1\times \cdots \times A_n\times X_{n+1}\times \cdots )=\prod _{k=1}^n{P}_k(A_k).$$

The finiteness is what makes this work: the "tail" factor ${\prod }_{k>n}{P}_k(X_k)=1$ costs nothing, whereas an infinite product of general measures diverges or vanishes. Existence again follows from Carathéodory applied to the algebra of cylinder sets, with the $\sigma$-additivity check supplied by a compactness argument.

This is the existence theorem for i.i.d. sequences. "Let $X_1,X_2,\dots$ be independent with law $\nu$" is not a definition until some measure space carries such a sequence; the product measure ${\nu }^{\otimes \mathbb{N}}$ on ${\mathbb{R}}^{\mathbb{N}}$, with $X_k$ the coordinate projections, is that space. The generalization to dependent coordinates specified by consistent finite-dimensional distributions is the Kolmogorov extension theorem (extension-theorem.md), of which this is the independent special case.

Note the limitation that survives to the general theorem: the product $\sigma$-algebra contains only sets determined by countably many coordinates. On ${\mathbb{R}}^{[0,\infty )}$ that excludes $\{\text{the path is continuous}\}$ — the technical obstruction that makes the construction of Brownian motion more than a corollary.

6. Where This Is Used

• Independence — independence is the statement that a joint law is a product measure.
• The Kolmogorov extension theorem — the dependent generalization of §5.
• Expectation — the layer-cake formula, and $E[XY]=E[X]E[Y]$ for independent $X,Y$ as an instance of Fubini.
• Distributions and the Schwartz space — convolution and mollification.