Measure Theory
1.1 Probability Spaces
A probability space is a triple \((\Omega, \mathcal{F}, P)\) where \(\Omega\) is a set of “outcomes,” \(\mathcal{F}\) is a set of “events,” and \(P : \mathcal{F} \to [0,1]\) is a function that assigns probabilities to events.
We assume that \(\mathcal{F}\) is a \(\sigma\)-field (or \(\sigma\)-algebra), i.e., a (nonempty) collection of subsets of \(\Omega\) that satisfy
if \(A \in \mathcal{F}\) then \(A^c \in \mathcal{F}\), and
if \(A_i \in \mathcal{F}\) is a countable sequence of sets then \(\bigcup_i A_i \in \mathcal{F}\).
A measure is a nonnegative countably additive set function; that is, a function \(\mu : \mathcal{F} \to \mathbb{R} \cup \{\infty\}\) with
\(\mu(A) \ge \mu(\emptyset) = 0\) for all \(A \in \mathcal{F}\), and
if \(A_i \in \mathcal{F}\) is a countable sequence of disjoint sets, then
\[ \mu\left(\bigcup_i A_i\right) = \sum_i \mu(A_i). \]
If \(\mu(\Omega) = 1\), we call \(\mu\) a probability measure. Probability measures are usually denoted by \(P\).
In general, a measure may take the value \(\infty\), i.e.,
\[ \mu : \mathcal{F} \to [0,\infty]. \]
Theorem 1.1 (Properties of Measures) Let \(\mu\) be a measure on \((\Omega, \mathcal{F})\) (a \(\sigma\)-field).
Monotonicity. If \(A \subset B\) then \(\mu(A) \le \mu(B)\).
Subadditivity. If \(A \subset \bigcup_{m=1}^{\infty} A_m\) then
\[ \mu(A) \le \sum_{m=1}^{\infty} \mu(A_m). \]
Continuity from below. If \(A_i \uparrow A\) (i.e., \(A_1 \subset A_2 \subset \cdots\) and \(\bigcup_i A_i = A\)), then \(\mu(A_i) \uparrow \mu(A)\).
Continuity from above. If \(A_i \downarrow A\) (i.e., \(A_1 \supset A_2 \supset \cdots\) and \(\bigcap_i A_i = A\)), with \(\mu(A_1) < \infty\), then \(\mu(A_i) \downarrow \mu(A)\).
The proof uses subtraction, so we must avoid the undefined expression \(\infty-\infty\). Moreover, the conclusion may fail without this condition.
For example, consider Lebesgue measure on \(\mathbb R\). Let
\[ A_n=[n,\infty)\downarrow\varnothing. \]
However,
\[ \mu(A_n)=\infty \quad\text{for every }n, \]
so \(\mu(A_n)\) does not converge to \(\mu(\varnothing)=0\).
Let \((X,\mathcal A,\mu)\) be a measure space. The following classes are not mutually exclusive.
Finite measure. A measure \(\mu\) is finite if \(\mu(X)<\infty\).
For example, Lebesgue measure on \([0,2]\) is finite, with total mass \(2\).
Probability measure. A measure \(\mu\) is a probability measure if \(\mu(X)=1\).
Thus, every probability measure is a finite measure. For example, Lebesgue measure on \([0,1]\) is a probability measure.
\(\sigma\)-finite measure. A measure \(\mu\) is \(\sigma\)-finite if there exist measurable sets \(E_1,E_2,\ldots\) such that
\[ \begin{gathered} X = \bigcup_{n=1}^{\infty} E_n, \\ \mu(E_n) < \infty \quad \text{for every } n. \end{gathered} \]
For example, Lebesgue measure on \(\mathbb R\) is \(\sigma\)-finite because
\[ \begin{gathered} \mathbb R = \bigcup_{n=1}^{\infty} [-n,n], \\ m([-n,n]) = 2n < \infty. \end{gathered} \]
Infinite measure. A measure \(\mu\) is infinite if \(\mu(X)=\infty\).
An infinite measure may or may not be \(\sigma\)-finite. Counting measure on \(\mathbb N\) is \(\sigma\)-finite, whereas counting measure on \(\mathbb R\) is not.
If we are given a set \(\Omega\) and a collection \(\mathcal{A}\) of subsets of \(\Omega\), then there is a smallest \(\sigma\)-field containing \(\mathcal{A}\). We call this the \(\sigma\)-field generated by \(\mathcal{A}\) and denote it by \(\sigma(\mathcal{A})\).
Let \(\mathbb{R}^d\) be the set of vectors \((x_1,\ldots,x_d)\) of real numbers and \(\mathcal{R}^d\) be the Borel sets, the smallest \(\sigma\)-field containing the open sets. When \(d=1\) we drop the superscript.
Measures on the real line. Measures on \((\mathbb{R},\mathcal{R})\) are defined by giving a Stieltjes measure function with the following properties:
\(F\) is nondecreasing.
\(F\) is right continuous, i.e. \(\lim_{y\downarrow x} F(y)=F(x)\).
Theorem 1.2 (Stieltjes Measures) Associated with each Stieltjes measure function \(F\) there is a unique measure \(\mu\) on \((\mathbb{R}, \mathcal{R})\) with
\[ \mu((a,b]) = F(b) - F(a) \]
When \(F(x) = x\) the resulting measure is called Lebesgue measure.
Corollary 1.1 (Generating the Borel σ-Field) Let \(\mathcal{C}(\mathbb{R}^d)\) be the collection of countable open rectangles of the form
\[ (a_1,b_1) \times \cdots \times (a_d,b_d), \]
where the endpoints satisfy
\[ a_i,b_i \in \mathbb{Q} \cup \{-\infty,+\infty\} \]
for \(i = 1,\ldots,d\).
Then \(\mathcal{C}(\mathbb{R}^d)\) generates the Borel \(\sigma\)-field on \(\mathbb{R}^d\); that is,
\[ \sigma\bigl(\mathcal{C}(\mathbb{R}^d)\bigr) = \mathcal{R}^d. \]
1.2 Random Variables
A function \(X : \Omega \to S\) is said to be a measurable map from \((\Omega, \mathcal{F})\) to \((S, \mathcal{S})\) if
\[ X^{-1}(B) \equiv \{\omega : X(\omega) \in B\} \in \mathcal{F} \]
for all \(B \in \mathcal{S}\).
If \((S, \mathcal{S}) = (\mathbb{R}^d, \mathcal{R}^d)\) and \(d > 1\), then \(X\) is called a random vector. Of course, if \(d = 1\), \(X\) is called a random variable.
We define \(\sigma(X)\) to be the smallest \(\sigma\)-algebra on \(\Omega\) such that \(X\) is measurable.
Theorem 1.3 (Measurability on a Generating Collection) Suppose \(\mathcal{A}\) generates \(\mathcal{S}\). If \(X^{-1}(A) \in \mathcal{F}\) for every \(A \in \mathcal{A}\), then \(X\) is a measurable map from \((\Omega, \mathcal{F})\) to \((S, \mathcal{S})\).
Theorem 1.4 (Composition of Measurable Maps) If the maps
\[ \begin{aligned} X &: (\Omega, \mathcal{F}) \to (S, \mathcal{S}), \\ f &: (S, \mathcal{S}) \to (T, \mathcal{T}) \end{aligned} \]
are measurable, then \(f(X)\) is a measurable map from \((\Omega, \mathcal{F})\) to \((T, \mathcal{T})\).
The Borel \(\sigma\)-field \(\mathcal{B}(\mathbb{R}^d)\) is the smallest \(\sigma\)-field that contains all open sets in \(\mathbb{R}^d\).
An important property is that every continuous function is Borel measurable.
Let \(f : \mathbb{R}^d \to \mathbb{R}\) be continuous. Then, for every \(B \in \mathcal{B}(\mathbb{R})\),
\[ f^{-1}(B) \in \mathcal{B}(\mathbb{R}^d). \]
More generally, if \(X\) is an \(\mathbb{R}^d\)-valued random vector and \(f\) is continuous, then \(f(X)\) is a random variable.
1.3 Distributions
The indicator function of a set \(A \in \mathcal{F}\):
\[ 1_A(\omega) = \begin{cases} 1, & \omega \in A, \\ 0, & \omega \notin A. \end{cases} \]
If \(X\) is a random variable, then \(X\) induces a probability measure on \(\mathbb{R}\), called its distribution by setting \(\mu(A) = P(X \in A)\) for Borel sets \(A\). The right-hand side can be written as \(P(X^{-1}(A))\).
The distribution of a random variable \(X\) is usually described by giving its distribution function,
\[ F(x) = P(X \le x). \]
Theorem 1.5 (Properties of Distribution Functions) Any distribution function \(F\) has the following properties:
\(F\) is nondecreasing.
The limits at infinity are
\[ \lim_{x\to\infty} F(x) = 1, \qquad \lim_{x\to-\infty} F(x) = 0. \]
\(F\) is right continuous, i.e., \(\lim_{y\downarrow x} F(y) = F(x)\).
If \(F(x-) = \lim_{y\uparrow x} F(y)\), then \(F(x-) = P(X < x)\).
The probability at a point is
\[ P(X = x) = F(x) - F(x-). \]
Theorem 1.6 (Existence of a Distribution) If \(F\) satisfies properties (i), (ii), and (iii), then \(F\) is the distribution function of some random variable.
1.4 Integration
1.4.1 Simple Functions
\(\varphi\) is said to be a simple function if
\[ \varphi(\omega) = \sum_{i=1}^n a_i 1_{A_i} \]
and \(A_i\) are disjoint sets with \(\mu(A_i) < \infty\). If \(\varphi\) is a simple function, we let
\[ \int \varphi \, d\mu = \sum_{i=1}^n a_i \mu(A_i). \]
Theorem 1.7 (Properties of the Integral) \(\varphi \ge \psi\) \(\mu\)-almost everywhere (or \(\varphi \ge \psi\) \(\mu\)-a.e.) means
\[ \mu(\{\omega : \varphi(\omega) < \psi(\omega)\}) = 0. \]
Let \(\varphi\) and \(\psi\) be simple functions.
Nonnegativity. If \(\varphi \ge 0\) a.e., then \(\int \varphi \, d\mu \ge 0\).
Homogeneity. For any \(a \in \mathbb{R}\), \(\int a\varphi \, d\mu = a \int \varphi \, d\mu\).
Additivity.
\[ \int (\varphi + \psi) \, d\mu = \int \varphi \, d\mu + \int \psi \, d\mu. \]
Monotonicity. If \(\varphi \le \psi\) a.e., then \(\int \varphi \, d\mu \le \int \psi \, d\mu\).
Equality almost everywhere. If \(\varphi = \psi\) a.e., then \(\int \varphi \, d\mu = \int \psi \, d\mu\).
Triangle inequality.
\[ \left|\int \varphi \, d\mu\right| \le \int |\varphi| \, d\mu. \]
1.4.2 Bounded Functions
Let \(E\) be a set with \(\mu(E) < \infty\) and let \(f\) be a bounded function that vanishes on \(E^c\).
To define the integral of \(f\), we observe that if \(\varphi, \psi\) are simple functions that have \(\varphi \le f \le \psi\), then we want to have
\[ \int \varphi \, d\mu \le \int f \, d\mu \le \int \psi \, d\mu, \]
so we let
\[ \begin{aligned} \int f \, d\mu &= \sup_{\varphi \le f} \int \varphi \, d\mu \\ &= \inf_{\psi \ge f} \int \psi \, d\mu. \end{aligned} \]
Corollary 1.2 (Properties of the Integral for Bounded Functions) Let \(E\) be a set with \(\mu(E) < \infty\). If \(f\) and \(g\) are bounded functions that vanish on \(E^c\), then:
Nonnegativity. If \(f \ge 0\) a.e., then \(\int f \, d\mu \ge 0\).
Homogeneity. For any \(a \in \mathbb{R}\), \(\int af \, d\mu = a \int f \, d\mu\).
Additivity.
\[ \int (f + g) \, d\mu = \int f \, d\mu + \int g \, d\mu. \]
Monotonicity. If \(g \le f\) a.e., then \(\int g \, d\mu \le \int f \, d\mu\).
Equality almost everywhere. If \(g = f\) a.e., then \(\int g \, d\mu = \int f \, d\mu\).
Triangle inequality.
\[ \left|\int f \, d\mu\right| \le \int |f| \, d\mu. \]
1.4.3 Nonnegative Functions
For \(f\ge 0\), define
\[ \int f\,d\mu = \sup_{h\in\mathcal{H}_f}\int h\,d\mu, \]
where \(\mathcal{H}_f\) consists of bounded functions \(h\) such that
\[ 0 \le h \le f, \qquad \mu(\{x : h(x) > 0\}) < \infty. \]
Lemma 1.1 (Approximation by Truncation) Let \(E_n \uparrow \Omega\) have \(\mu(E_n) < \infty\) and let \(a \wedge b = \min(a,b)\). Then
\[ \int_{E_n} f \wedge n\, d\mu \uparrow \int f\, d\mu \qquad \text{as } n \uparrow \infty. \]
Corollary 1.3 (Properties of the Integral for Nonnegative Functions) Suppose \(f,g \ge 0\).
Nonnegativity. \(\int f \, d\mu \ge 0\).
Homogeneity. If \(a > 0\), then \(\int af \, d\mu = a \int f \, d\mu\).
Additivity.
\[ \int (f + g) \, d\mu = \int f \, d\mu + \int g \, d\mu. \]
Monotonicity. If \(0 \le g \le f\) a.e., then \(\int g \, d\mu \le \int f \, d\mu\).
Equality almost everywhere. If \(0 \le g = f\) a.e., then \(\int g \, d\mu = \int f \, d\mu\).
Let \((X,\mathcal A,\mu)\) be a measure space. For a nonnegative measurable function \(f:X\to[0,\infty]\), define
\[ \int_X f \, d\mu = \sup_{\substack{s\text{ nonnegative simple} \\ s \le f}} \int_X s \, d\mu. \]
In other words, approximate \(f\) from below using nonnegative simple functions and take the supremum of their integrals. The integral is allowed to equal \(+\infty\).
1.4.4 General Functions
We say \(f\) is integrable if
\[ \int |f|\,d\mu < \infty. \]
The positive and negative parts of \(f\) are
\[ \begin{aligned} f^+(x) &= f(x)\vee 0, \\ f^-(x) &= (-f(x))\vee 0, \end{aligned} \]
where \(a\vee b=\max(a,b)\). They satisfy
\[ \begin{aligned} f(x) &= f^+(x)-f^-(x), \\ |f(x)| &= f^+(x)+f^-(x). \end{aligned} \]
We define the integral of \(f\) by
\[ \int f \, d\mu = \int f^+ \, d\mu - \int f^- \, d\mu. \]
Lemma 1.2 (Integral of a Difference) If \(f=f_1-f_2\), where \(f_1,f_2\ge 0\) and \(\int f_i\,d\mu<\infty\) for \(i=1,2\), then
\[ \int f\,d\mu = \int f_1\,d\mu - \int f_2\,d\mu. \]
Theorem 1.8 (Properties of the Integral for General Functions) Suppose \(f\) and \(g\) are integrable.
Nonnegativity. If \(f \ge 0\) a.e., then \(\int f \, d\mu \ge 0\).
Homogeneity. For all \(a \in \mathbb{R}\), \(\int af \, d\mu = a \int f \, d\mu\).
Additivity.
\[ \int (f + g) \, d\mu = \int f \, d\mu + \int g \, d\mu. \]
Monotonicity. If \(g \le f\) a.e., then \(\int g \, d\mu \le \int f \, d\mu\).
Equality almost everywhere. If \(g = f\) a.e., then \(\int g \, d\mu = \int f \, d\mu\).
Triangle inequality.
\[ \left|\int f \, d\mu\right| \le \int |f| \, d\mu. \]
1.4.5 Notation for Special Cases
The notation for an integral depends on the underlying measure:
Lebesgue measure. For \(\lambda\) on \((\mathbb{R}^d, \mathcal{R}^d)\),
\[ \int f(x) \, dx = \int f \, d\lambda. \]
Lebesgue measure on an interval. For \(E = [a,b] \subset \mathbb{R}\),
\[ \int_a^b f(x) \, dx = \int_E f \, d\lambda. \]
Stieltjes measure. For \(\mu\) on \((\mathbb{R}, \mathcal{R})\), with \(\mu((a,b]) = G(b) - G(a)\) for \(a < b\),
\[ \int f(x) \, dG(x) = \int f \, d\mu. \]
Counting measure. For a countable set \(\Omega\), with \(\mathcal{F}\) containing all subsets,
\[ \sum_{i \in \Omega} f(i) = \int f \, d\mu. \]
1.5 Properties of the Integral
1.5.1 Radon–Nikodym Derivatives
Theorem 1.9 (Radon–Nikodym Theorem) Let \(\mu\) and \(\nu\) be \(\sigma\)-finite measures on the same measurable space \((\Omega,\mathcal F)\). If
\[ \nu\ll\mu, \]
meaning
\[ \mu(A)=0\implies\nu(A)=0, \]
then there exists a nonnegative measurable function \(f\) such that
\[ \nu(A)=\int_A f\,d\mu \]
for every \(A\in\mathcal F\).
The function \(f\) is unique up to a \(\mu\)-null set and is called the Radon–Nikodym derivative:
\[ f=\frac{d\nu}{d\mu}. \]
Let \(X\) be a real-valued random variable with distribution \(\mu_X(A)=P(X\in A)\) for Borel sets \(A\), and let \(\lambda\) denote Lebesgue measure on \(\mathbb R\).
Probability density function (PDF): If \(\mu_X\ll\lambda\), the Radon–Nikodym derivative
\[ f_X=\frac{d\mu_X}{d\lambda} \]
is called a probability density function of \(X\). Thus, for every Borel set \(A\),
\[ P(X\in A)=\int_A f_X\,d\lambda=\int_A f_X(x)\,dx. \]
The density is nonnegative, unique up to a \(\lambda\)-null set, and satisfies
\[ \int_{\mathbb R}f_X(x)\,dx=1. \]
1.5.2 Integral Inequalities
Theorem 1.10 (Jensen’s Inequality) Suppose \(\varphi\) is convex, that is,
\[ \lambda\varphi(x) + (1-\lambda)\varphi(y) \ge \varphi\bigl(\lambda x + (1-\lambda)y\bigr) \]
for all \(\lambda \in (0,1)\) and \(x,y \in \mathbb{R}\). If \(\mu\) is a probability measure, and \(f\) and \(\varphi(f)\) are integrable, then
\[ \varphi\left(\int f \, d\mu\right) \le \int \varphi(f) \, d\mu. \]
For \(1 \le p < \infty\), let
\[ \|f\|_p = \left(\int |f|^p \, d\mu\right)^{1/p}. \]
Notice that \(\|cf\|_p = |c| \cdot \|f\|_p\) for any real number \(c\).
Theorem 1.11 (Hölder’s Inequality) If \(p,q \in (1,\infty)\) with
\[ \frac{1}{p} + \frac{1}{q} = 1, \]
then
\[ \int |fg| \, d\mu \le \|f\|_p \|g\|_q. \]
Corollary 1.4 (Cauchy–Schwarz Inequality) If \(p=q=2\), then Hölder’s inequality becomes the Cauchy–Schwarz inequality:
\[ \int |fg| \, d\mu \le \|f\|_2 \|g\|_2. \]
1.5.3 Convergence Theorems
Let \((X,\mathcal A,\mu)\) be a measure space, and let \(f_n,f:X\to\mathbb R\) be measurable functions.
Convergence in measure. We say that \(f_n\) converges to \(f\) in measure if, for every \(\epsilon>0\), \[ \mu\bigl(\{x\in X:|f_n(x)-f(x)|>\epsilon\}\bigr)\to0 \] as \(n\to\infty\).
Almost-everywhere convergence. We say that \(f_n\) converges to \(f\) almost everywhere if there exists a measurable set \(Z\subseteq X\) with \(\mu(Z)=0\) such that \[ \lim_{n\to\infty}f_n(x)=f(x) \] for every \(x\in X\setminus Z\). We write \[ f_n\xrightarrow{\mathrm{a.e.}}f. \]
Uniform convergence. We say that \(f_n\) converges to \(f\) uniformly on \(X\) if, for every \(\epsilon>0\), there exists an integer \(N\ge1\) such that \[ |f_n(x)-f(x)|<\epsilon \] for every \(n\ge N\) and every \(x\in X\). The integer \(N\) may depend on \(\epsilon\), but not on \(x\).
Almost-uniform convergence. We say that \(f_n\) converges to \(f\) almost uniformly on \(X\) if, for every \(\delta>0\), there exists a measurable set \(D_\delta\subseteq X\) such that \[ \mu(D_\delta)<\delta \] and \(f_n\) converges to \(f\) uniformly on \(X\setminus D_\delta\).
The exceptional set \(D_\delta\) may depend on \(\delta\), but not on \(n\).
On a space of finite measure, almost-everywhere convergence implies convergence in measure, but the converse need not hold.
For measurable functions on a measure space \((X,\mathcal F,\mu)\), we have \[ \text{uniform convergence} \;\Longrightarrow\; \text{almost-uniform convergence} \;\Longrightarrow\; \text{almost-everywhere convergence}, \] and \[ \text{almost-uniform convergence} \;\Longrightarrow\; \text{convergence in measure}. \]
If \(\mu(X)<\infty\), Egorov’s theorem also gives \[ \text{almost-everywhere convergence (almost surely)} \;\Longrightarrow\; \text{almost-uniform convergence} \;\Longrightarrow\; \text{convergence in measure (convergence in probability)}. \] The finite-measure assumption cannot generally be omitted from this implication.
Theorem 1.12 (Bounded Convergence Theorem) Let \(E\) be a set with \(\mu(E)<\infty\). Suppose \(f_n\) vanishes on \(E^c\), \(|f_n(x)|\le M\), and \(f_n\to f\) in measure. Then
\[ \int f \, d\mu = \lim_{n\to\infty} \int f_n \, d\mu. \]
Theorem 1.13 (Fatou’s Lemma) If \(f_n \ge 0\) then
\[ \liminf_{n\to\infty}\int f_n\,d\mu \ge \int\left(\liminf_{n\to\infty}f_n\right)\,d\mu. \]
Theorem 1.14 (Monotone Convergence Theorem) If \(f_n\ge0\) and \(f_n\uparrow f\), then
\[ \int f_n\,d\mu \uparrow \int f\,d\mu. \]
Theorem 1.15 (Dominated Convergence Theorem) If \(f_n\to f\) a.e., \(|f_n|\le g\) for all \(n\), and \(g\) is integrable, then
\[ \int f_n\,d\mu \longrightarrow \int f\,d\mu. \]
Although the versions of Theorem 1.13, Theorem 1.14, and Theorem 1.15 presented in class assume that \(\mu\) is \(\sigma\)-finite, all three theorems hold for arbitrary measures and can be proved without this assumption using other methods.
1.6 Expected Value
We now specialize to integration with respect to a probability measure \(P\).
If \(X\) is a random variable on \((\Omega,\mathcal F,P)\), then we define its expected value to be \[ EX=\int X\,dP, \]
if the integral exists. This is the case if \(X\) is integrable, i.e., if \(EX<\infty\).
When \(X \ge 0\), which always makes sense, but may be \(\infty\).
Theorem 1.16 (Expected Value of a Function of a Random Variable) Let \(X\) be a random element of \((S,\mathcal S)\) with distribution \(\mu\), i.e., \(\mu(A)=P(X\in A)\). If \(f\) is a measurable function from \((S,\mathcal S)\) to \((\mathbb R,\mathcal R)\) such that \(f\ge0\) or \(E|f(X)|<\infty\), then \[ Ef(X)=\int_S f(y)\,\mu(dy) = \int_S f(y) f_X(y)\,dy. \]
1.7 Product Measures
Let \((X,\mathcal A,\mu_1)\) and \((Y,\mathcal B,\mu_2)\) be two \(\sigma\)-finite measure spaces. Let
\[ \begin{aligned} \Omega&=X\times Y=\{(x,y):x\in X,\ y\in Y\},\\ \mathcal S&=\{A\times B:A\in\mathcal A,\ B\in\mathcal B\}. \end{aligned} \]
Sets in \(\mathcal S\) are called rectangles. It is easy to see that \(\mathcal S\) is a semi-algebra:
\[ (A\times B)\cap(C\times D)=(A\cap C)\times(B\cap D), \]
\[ (A\times B)^c =(A^c\times B)\cup(A\times B^c)\cup(A^c\times B^c). \]
Let \(\mathcal F=\mathcal A\times\mathcal B\) be the \(\sigma\)-algebra generated by \(\mathcal S\).
Theorem 1.17 (Existence and Uniqueness of Product Measures) There exists a unique measure \(\mu\) on \(\mathcal F\) such that
\[ \mu(A\times B)=\mu_1(A)\mu_2(B) \]
for every \(A\in\mathcal A\) and \(B\in\mathcal B\). This measure is called the product measure and is denoted by \(\mu_1\times\mu_2\).
Theorem 1.18 (Fubini’s Theorem) Let \(\mu=\mu_1\times\mu_2\), and let \(f:X\times Y\to\mathbb R\) be \(\mathcal F\)-measurable. If \(f\ge0\) or
\[ \int_{X\times Y}|f|\,d\mu<\infty, \]
then
\[ \begin{aligned} \int_{X\times Y}f\,d\mu &=\int_X\left(\int_Y f(x,y)\,\mu_2(dy)\right)\mu_1(dx)\\ &=\int_Y\left(\int_X f(x,y)\,\mu_1(dx)\right)\mu_2(dy). \end{aligned} \]