Product Measures and Change of Variables

Two computations dominate the rest of this guidebook's analysis. One is splitting an integral over a product space into iterated integrals: integrate over $x$, then over $y$. The other is changing variables, as when an integral over $\mathbb{R}^n$ is done in polar coordinates, or a metric is pulled back by a diffeomorphism. Course 1 did both by computation (1A.9 Multiple Integrals and Change of Variables). This chapter says when they are legitimate. The answers are Tonelli's and Fubini's theorems, which say iterated integrals may be taken in either order provided the integrand is non-negative or absolutely integrable, and the change-of-variables formula, in which the Jacobian determinant records how a map stretches volume.

With those in hand we compute the most important integral in the guidebook:

It normalises the heat kernel (6A.3 The Heat Equation on ℝⁿ), and it normalises Perelman's weighted measure (12A.3 The 𝓦-Entropy), in which flat space is the case of equality. Along the way come the volumes of balls in every dimension, which turn out to have a surprise in store.

By the end of this chapter you will be able to:

• state Tonelli's and Fubini's theorems and check their hypotheses, with counterexamples when they fail;
• use the change-of-variables formula, including polar and spherical coordinates;
• compute $\int e^{-x^2}dx = \sqrt\pi$ and its $n$-dimensional and scaled versions;
• compute the volume of the unit ball and the area of the unit sphere in $\mathbb{R}^n$;
• use the scaling rule $dx \mapsto \lambda^n\,dx$ to see how integrals behave under dilation.

5.1Seeing inside by integrating along lines

5.2Product measures

Let $(X, \mathcal{B}_X, \mu)$ and $(Y, \mathcal{B}_Y, \nu)$ be measure spaces. A measurable rectangle is a set $A \times B$ with $A \in \mathcal{B}_X$, $B \in \mathcal{B}_Y$. The product σ-algebra $\mathcal{B}_X \times \mathcal{B}_Y$ is the smallest σ-algebra containing the measurable rectangles.

Tao builds it in §1.7 by the Carathéodory extension theorem: define the measure on finite unions of rectangles, check countable additivity there, and extend. For Lebesgue measure, $m^{d_1} \times m^{d_2}$ agrees with $m^{d_1 + d_2}$ on Borel sets, and $m^{d_1+d_2}$ is its completion (add all subsets of null sets).

5.3Tonelli and Fubini

Architecture of the proofs (Tao, §1.7.4). For $f = 1_{A\times B}$ both iterated integrals equal $\mu(A)\nu(B)$. The class of sets $E$ for which the theorem holds for $1_E$ is closed under increasing unions (monotone convergence) and under differences of nested sets of finite measure (subtraction), and it contains finite unions of rectangles; a "monotone class" argument then shows it contains the whole product σ-algebra. From indicators, Tonelli follows by simple functions and monotone convergence, the pattern of 3A.4 Measures, Probability and Weights. Fubini follows by applying Tonelli to $f^+$ and $f^-$, which are both integrable because Tonelli applied to $|f|$ says so.

In practice: to exchange the order of integration, check that $\iint|f| < \infty$ by computing it in either order, which Tonelli always allows. Then Fubini permits any order for $f$ itself.

5.3.1When the order matters

Both hypotheses are needed.

5.3.2Slicing

Applied to an indicator function, Tonelli says the measure of a set is the integral of the measures of its slices: if $E \subseteq \mathbb{R}^{n+1}$ and $E_t = \{x : (x, t) \in E\}$, then $m(E) = \int m(E_t)\,dt$. This is Cavalieri's principle: solids with slices of equal area at every height have equal volume. It gives, for example, the volume of a solid of revolution, and it gives the layer-cake formula for an integral, Lebesgue's horizontal slicing (3A.3 The Lebesgue Integral) made exact:

by applying Tonelli to the region under the graph of $f$ (Exercise 5.10).

5.4Change of variables

The derivative of a map is its best linear approximation (2B.8 Calculus in Several Variables), so the first case to understand is a linear map.

Architecture of the proof (Munkres, "Changing Variables"; Stein–Shakarchi, chapter 2). Near a point $x_0$, $\phi$ is approximately the affine map $x \mapsto \phi(x_0) + D\phi(x_0)(x - x_0)$, which by Proposition 5.6 maps a small cube $Q$ to a parallelepiped of volume $|\det D\phi(x_0)|\,m(Q)$. Uniform continuity of $D\phi$ on compact sets makes the error small relative to $m(Q)$, uniformly over a fine grid of cubes. Summing over the grid proves the formula for indicators of cubes, then of open sets, and then for all non-negative measurable $f$ by the usual monotone-class and simple-function route.

The factor $|\det D\phi|$ is the Jacobian. In one variable it is the $|\phi'|$ of 2A.11 The Riemann Integral's substitution rule. On a manifold it becomes the way the volume form changes between charts (8A.8 Differential Forms and Stokes’ Theorem), and under a flow of diffeomorphisms it is the divergence that measures the rate of change of volume (8A.6 Flows and the Lie Derivative).

5.4.1Polar coordinates

In the plane, $\phi(r, \theta) = (r\cos\theta, r\sin\theta)$ maps $(0, \infty) \times (0, 2\pi)$ onto the plane minus a ray (a null set), with $\det D\phi = r$. So

In $\mathbb{R}^n$, every non-zero $x$ is $r\omega$ with $r = |x| > 0$ and $\omega \in S^{n-1}$, and Lebesgue measure splits as

where $\sigma$ is surface measure on the unit sphere. For a radial function $f(x) = g(|x|)$ this gives

with $|S^{n-1}| = \sigma(S^{n-1})$ the area of the unit sphere. The factor $r^{n-1}$ says that a thin shell of radius $r$ and thickness $dr$ has volume $|S^{n-1}|r^{n-1}dr$: surface area times thickness (Figure 5.2). For the purposes of this guidebook, $(*)$ can be taken as the definition of $|S^{n-1}|$ (surface measure is constructed properly on manifolds in 8A.8 Differential Forms and Stokes’ Theorem); its value is computed below.

5.5The Gaussian integral

The trick of squaring a one-dimensional integral to make it two-dimensional, where polar coordinates apply, is attributed to Poisson. It works only because Tonelli permits the exchange.

5.5.1Volumes of balls and spheres

Apply $(*)$ to $e^{-|x|^2}$:

where $\Gamma(s) = \int_0^\infty t^{s-1}e^{-t}\,dt$ is Euler's gamma function (substitute $t = r^2$), which satisfies $\Gamma(s+1) = s\Gamma(s)$, $\Gamma(1) = 1$ and $\Gamma(\tfrac12) = \sqrt\pi$. So

Check: $\omega_2 = \pi$, $\omega_3 = \frac{\pi^{3/2}}{\Gamma(5/2)} = \frac{\pi^{3/2}}{\frac34\sqrt\pi} = \frac43\pi$, and $|S^2| = 4\pi$.

5.5.2Scaling

A special case of change of variables is used so often it deserves its own line. For $\lambda > 0$,

Combined with how derivatives scale, $\nabla[f(\lambda x)] = \lambda(\nabla f)(\lambda x)$, it predicts the exponents in every inequality that is invariant under dilation. For instance, $\int|\nabla u|^2$ and $\big(\int|u|^p\big)^{2/p}$ scale the same way under $u(x) \mapsto u(\lambda x)$ exactly when $p = \frac{2n}{n-2}$, which is the exponent in the Sobolev inequality of 4A.10 Sobolev Embeddings and Critical Exponents. This kind of dimensional analysis is thread S, and it is how one sees that Perelman's noncollapsing estimate $\mathrm{vol}\,B(x, r) \geq \kappa r^n$ is scale-invariant (3A.1 The Problem of Measure, 12A.4 κ-Noncollapsing).

5.6History

Bonaventura Cavalieri published his principle of comparing solids by slices in 1635. Guido Fubini proved his theorem for Lebesgue integrals in 1907, and Leonida Tonelli the version for non-negative functions in 1909. Carl Jacobi studied the functional determinants that bear his name in 1841. The value $\sqrt\pi$ of the Gaussian integral goes back to de Moivre and Laplace in the 18th century; Gauss used the bell curve for errors of observation in 1809. Johann Radon's paper on recovering a function from its line integrals appeared in 1917, and computed tomography put it to clinical use in the 1970s.

5.7Exercises

Solution

(b) The inner integral in $y$ is $\big[\frac{y}{x^2+y^2}\big]_0^1 = \frac{1}{x^2 + 1}$, whose integral over $[0, 1]$ is $\frac\pi4$. By the antisymmetry $f(y, x) = -f(x, y)$, the other order gives $-\frac\pi4$. The function is not absolutely integrable near the origin: in polar coordinates $|f| = \frac{|\cos 2\theta|}{r^2}$, and $\int\frac{1}{r^2}\,r\,dr$ diverges at $0$.

Solution

The ellipse is the image of the unit disc under $\mathrm{diag}(a, b)$. For the ellipsoid, $x = A^{-1/2}y$ maps the unit ball onto it, and $\det A^{-1/2} = (\det A)^{-1/2}$.

Solution

$\Gamma(n/2 + 1) = \frac n2\Gamma(n/2)$, so $\omega_n/\omega_{n-2} = \pi/\frac n2 = \frac{2\pi}{n}$, which is at least $1$ exactly when $n \leq 2\pi$, that is, $n \leq 6$. Along each of the even and odd chains, $\omega_n$ is a product of factors $\frac{2\pi}{k}$, which tend to $0$, so $\omega_n \to 0$.

Solution

(a) The integrand is odd in $x_i$. (b) For $i \neq j$, odd again. For $i = j$, by Tonelli the other coordinates integrate to $1$, and $\int x^2(4\pi\tau)^{-1/2}e^{-x^2/4\tau}dx = 2\tau$ (substitute $x = 2\sqrt\tau y$ and use $\int y^2e^{-y^2} = \frac{\sqrt\pi}2$, 3A.3 The Lebesgue Integral). (c) $\nabla f = \frac{x}{2\tau}$, so $\tau|\nabla f|^2 = \frac{|x|^2}{4\tau} = f$, and the integrand is $2f - n$. Its integral is $\frac{2}{4\tau}\cdot2n\tau - n = 0$.