V. Product spaces and product measures

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Find these notes at gregorkb.github.io/probtheory/

Product spaces

• Product measurable space:Let \((\Omega_1,\mathcal{F}_1)\) and \((\Omega_2,\mathcal{F}_2)\) be measurable spaces. The define:

  • \(\Omega_1 \times \Omega_2 = \{(\omega_1,\omega_2): \omega_1 \in \Omega_1, \omega_2 \in \Omega_2\}\) as the Cartesian product of \(\Omega_1\) and \(\Omega_2\).

  • For \(A_1 \in \mathcal{F}_1\) and \(A_2 \in \mathcal{F}_2\), call \(A_1 \times A_2 = \{(\omega_1,\omega_2) : \omega_1 \in A_1, \omega_2 \in A_2\}\) a measurable rectangle.

  • \(\mathcal{F}_1 \times \mathcal{F}_2 = \sigma \langle \{ A_1 \times A_2: A_1 \in \mathcal{F}_1, A_2 \in \mathcal{F}_2\} \rangle\) is the product \(\sigma\)-algebra of \(\mathcal{F}_1\) and \(\mathcal{F}_2\).

  • \((\Omega_1 \times \Omega_2, \mathcal{F}_1 \times \mathcal{F}_2)\) is the product measurable space.

• Class of measurable rectangles is a semi-algebra: Given measurable spaces \((\Omega_1,\mathcal{F}_1)\) and \((\Omega_2,\mathcal{F}_2)\), Show that the collection \[ \mathcal{R}\equiv \{A_1 \times A_2: A_1 \in \mathcal{F}_1, A_2 \in \mathcal{F}_2\} \] of measurable rectangles is a semi-algebra on \(\Omega_1 \times \Omega_2\). See page 144 of Resnick.

• Cross sections of sets and functions: Let \((\Omega_1 \times \Omega_2, \mathcal{F}_1 \times \mathcal{F}_2)\) be a product measurable space.

  • Given a set \(A \in \mathcal{F}_1 \times \mathcal{F}_2\) and \(\omega_1 \in \Omega_1\), define \(A_{1\omega_1} = \{\omega_2 \in \Omega_2: (\omega_1,\omega_2) \in A\}\) as the \(\omega_1\)-section of \(A\). Likewise, for each \(\omega_2 \in \Omega_2\) define \(A_{2\omega_2} = \{\omega_1 \in \Omega_1: (\omega_1,\omega_2) \in A\}\) as the \(\omega_2\)-section of \(A\).

  • Given another measurable space \((\Omega_3,\mathcal{F}_3)\) and a function \(f:\Omega_1\times \Omega_2 \to \Omega_3\) which is \(\langle \mathcal{F}_1 \times \mathcal{F}_2,\mathcal{F}_3 \rangle\)-measurable, define \(f_{1\omega_1}(\omega_2) = f(\omega_1,\omega_2)\) for all \(\omega_2 \in \Omega_2\) as the \(\omega_1\)-section of \(f\). Likewise define \(f_{2\omega_2}(\omega_1) = f(\omega_1,\omega_2)\) for all \(\omega_1 \in \Omega_1\) as the \(\omega_2\)-section of \(f\).

• Measurability of cross sections: Let \((\Omega_1 \times \Omega_2, \mathcal{F}_1 \times \mathcal{F}_2)\) and \((\Omega_3,\mathcal{F}_3)\) be measurable spaces and \(f:\Omega_1\times \Omega_2 \to \Omega_3\) be a \(\langle \mathcal{F}_1\times \mathcal{F}_2, \mathcal{F}_3\rangle\)-measurable function.

  • Show that \(A_{1\omega_1} \in \mathcal{F}_2\) for all \(\omega_1 \in \Omega_1\), whereby we also have \(A_{2\omega_2} \in \mathcal{F}_1\) for all \(\omega_2 \in \Omega_2\).

  • Show that \(f_{1\omega_1}\) is \(\langle \mathcal{F}_2,\mathcal{F}_3\rangle\)-measurable for all \(\omega_1 \in \Omega_1\), whereby we also have that \(f_{2\omega_2}\) is \(\langle \mathcal{F}_1,\mathcal{F}_3\rangle\)-measurable for all \(\omega_2 \in \Omega_2\).

  See Proposition 5.1.1 on page 148 of A&L, Section 5.7 starting on page 145 of Resnick.

• Building a product measure: Let \((\Omega_1 \times \Omega_2,\mathcal{F}_1\times \mathcal{F}_2)\) be a product measurable space and let \(\mu_1\) and \(\mu_2\) be finite measures on \((\Omega_1,\mathcal{F}_1)\) and \((\Omega_2,\mathcal{F}_2)\), respectively. Moreover, define the set functions \[ \begin{align} \mu_{12}(A) &= \int_{\Omega_1} \mu_2(A_{1\omega_1})d\mu(\omega_1)\\ \mu_{21}(A) &= \int_{\Omega_2} \mu_1(A_{2\omega_2})d\mu(\omega_2) \end{align} \] for all \(A \in \mathcal{F}_1\times \mathcal{F}_2\).

  • Show that for each \(A \in \mathcal{F}_1\times \mathcal{F}_2\) the function \(\mu_2(A_{1\omega_1})\) for \(\omega_1 \in \Omega_1\) is \(\langle \mathcal{F}_1,\mathcal{B}(\mathbb{R})\rangle\)-measurable, whereby we also have that \(\mu_1(A_{2\omega_2})\) for \(\omega_2 \in \Omega_2\) is a \(\langle \mathcal{F}_2,\mathcal{B}(\mathbb{R})\rangle\)-measurable function.

  • Show that \(\mu_{12}\) is a measure on \(\mathcal{F}_1\times\mathcal{F}_2\), whereby \(\mu_{21}\) is also a measure on \(\mathcal{F}_1\times\mathcal{F}_2\).

  • Show that \(\mu_{12}(A) = \mu_{21}(A)\) for all \(A \in \mathcal{F}_1\times \mathcal{F}_2\).

  The measure \(\mu \equiv \mu_{12} = \mu_{21}\), which we may also denote as \(\mu_1 \times \mu_2\), is called the product measure on \(\mathcal{F}_1 \times \mathcal{F}_2\). The triple \((\Omega_1\times \Omega_2, \mathcal{F}_1\times \mathcal{F}_2,\mu_1\times \mu_2)\) is called the product measure space.

  See Theorem 5.1.2 on page 149 of A&L and Definition 5.1.3 on page 151 ibid. There it is only assumed that \(\mu_1\) and \(\mu_2\) are \(\sigma\)-finite, so the result stated here is less general (but easier to prove).

Product measures and integration

• Tonelli’s Theorem: Let \((\Omega_1,\mathcal{F}_1,\mu_1)\) and \((\Omega_2,\mathcal{F}_2,\mu_2)\) be \(\sigma\)-finite measure spaces and let \(f: \Omega_1\times \Omega_2 \to [0,\infty]\) be a non-negative \(\langle \mathcal{F}_1\times\mathcal{F}_2,\mathcal{B}(\mathbb{R})\rangle\)-measurable function. Define the functions \[ g_1(\omega_1) = \int_{\Omega_2} f_{1\omega_1}d \mu_2, \quad \omega_1 \in \Omega_1 \] and \[ g_2(\omega_2) = \int_{\Omega_1} f_{2\omega_2}d\mu_1, \quad \omega_2 \in \Omega_2. \]

  • Show that \(g_1(\omega_1)\) and \(g_2(\omega_2)\) are \(\langle \mathcal{F}_1,\mathcal{B}(\bar{\mathbb{R}}) \rangle\)- and \(\langle\mathcal{F}_2,\mathcal{B}(\bar{\mathbb{R}}) \rangle\)-measurable functions, respectively.

  • Show that \(\int_{\Omega_1 \times\Omega_2} f d\mu = \int_{\Omega_1} g_1 d \mu_1 = \int_{\Omega_2} g_2 d \mu_2\), where \(\mu = (\mu_1 \times \mu_2)\).

  See Theorem 5.2.1 on page 152 of A&L.

• Fubini’s Theorem: Let \((\Omega_1,\mathcal{F}_1,\mu_1)\) and \((\Omega_2,\mathcal{F}_2,\mu_2)\) be \(\sigma\)-finite measure spaces and \(f \in L^1(\Omega_1\times \Omega_2,\mathcal{F}_1\times \mathcal{F}_2,\mu_1 \times \mu_2)\). Show that there exists a set \(B_1 \in \mathcal{F}_1\) such that

  1. \(\mu_1(\Omega_1\setminus B_1) = 0\)

  2. \(f_{1\omega_1} \in L^1(\Omega_2,\mathcal{F}_2,\mu_2)\) for each \(\omega_1 \in B_1\).

  3. The function \[ g_1(\omega_1) \equiv \left\{\begin{array}{ll} \int_{\Omega_2} f_{1\omega_1}d \mu_2,& \omega_1 \in B\\ 0,& \omega_1 \notin B \end{array}\right. \] is \(\langle \mathcal{F}_1,\mathcal{B}(\mathbb{R}) \rangle\)-measurable and \[ \int_{\Omega_1} g_1 d \mu_1 = \int_{\Omega_1 \times \Omega_2} f d(\mu_1 \times \mu_2). \]

  See Theorem 5.2.2 on page 153 of A&L.

• Integration by parts: Let \(F_1\) and \(F_2\) be non-decreasing, right-continuous functions on \([a,b]\) with no common points of discontinuity in \((a,b]\). Let \(dF_1\) and \(dF_2\) denote integration with respect to the Lebesgue-Stieltjes measures \(\mu_{F_1}\) and \(\mu_{F_2}\) defined with \(F_1\) and \(F_2\), respectively. Show that \[ \int_{(a,b]} F_1(x) dF_2(x) = F_1(b)F_2(b) - F_1(a)F_2(a) + \int_{(a,b]}F_2(x) dF_1(x). \] See Theorem 5.2.3 on page 155 of A&L.