Product Measures

Last modified: 2026-10-31 15:22:31 (PDT)

1 Product measures

Definition 1 (Measurable rectangle) Let \((S, \mathcal{S})\) and \((T, \mathcal{T})\) be measurable spaces. A measurable rectangle is a Cartesian product \(A \times B\) with \(A \in \mathcal{S}\) and \(B \in \mathcal{T}\).

Definition 2 (Product \(\sigma\)-algebra) For the measurable spaces \((S, \mathcal{S})\) and \((T, \mathcal{T})\) of Definition 1, the product \(\sigma\)-algebra \(\mathcal{S}\otimes \mathcal{T}\) is the \(\sigma\)-algebra on \(S \times T\) generated by the measurable rectangles.

Example 1 (Product \(\sigma\)-algebras)  

• Give the die rolls \(D\) the \(\sigma\)-algebra \(\mathcal{S}\) of all its subsets. Each single pair \({\mathopen{}\left\{(a, b)\right\}\mathclose{}} = {\mathopen{}\left\{a\right\}\mathclose{}} \times {\mathopen{}\left\{b\right\}\mathclose{}}\) in \(D \times D\) is a measurable rectangle. Every subset of \(D \times D\) is a union of finitely many single pairs, and \(\mathcal{S}\otimes \mathcal{S}\) contains finite unions of its sets (Theorem 1 in Sigma-Algebras), as well as \(\emptyset\), so \(\mathcal{S}\otimes \mathcal{S}\) is the collection of all subsets of \(D \times D\).
• The rectangle \([0, 2] \times [0, 3]\) is a measurable rectangle for the Borel \(\sigma\)-algebras, so it is in \(\mathcal{B}\otimes \mathcal{B}\).

Definition 3 (Product measure) Let \((S, \mathcal{S}, \mu)\) and \((T, \mathcal{T}, \nu)\) be measure spaces with \(\sigma\)-finite measures \(\mu\) and \(\nu\). The product measure \(\mu\otimes \nu\) is the measure on the product \(\sigma\)-algebra \(\mathcal{S}\otimes \mathcal{T}\) that gives each measurable rectangle the product of the measures of its sides:

\[(\mu\otimes \nu)(A \times B) = \mu(A) \, \nu(B) \quad \text{for all } A \in \mathcal{S}\text{ and } B \in \mathcal{T},\]

with \(0 \cdot \infty = \infty \cdot 0 \stackrel{\text{def}}{=}0\) and \(x \cdot \infty = \infty \cdot x \stackrel{\text{def}}{=}\infty\) for \(x > 0\). Exactly one measure has these values [1], Theorem 18.2.

Example 2 (Product measures)  

• For the counting measure \(\mu\) on the die rolls \(D\), \((\mu\otimes \mu)(A \times B) = \mathopen{}\left|A\right|\mathclose{} \mathopen{}\left|B\right|\mathclose{}\), the number of pairs in \(A \times B\). For example, the set of pairs in which both rolls are even is \({\mathopen{}\left\{2, 4, 6\right\}\mathclose{}} \times {\mathopen{}\left\{2, 4, 6\right\}\mathclose{}}\), with

  \[ \begin{aligned} (\mu\otimes \mu)({\mathopen{}\left\{2, 4, 6\right\}\mathclose{}} \times {\mathopen{}\left\{2, 4, 6\right\}\mathclose{}}) &= 3 \cdot 3 \\ &= 9 \end{aligned} \]

  pairs.

• For Lebesgue measure \(\lambda\) on \(\mathbb{R}\),

  \[ \begin{aligned} (\lambda\otimes \lambda)([0, 2] \times [0, 3]) &= 2 \cdot 3 \\ &= 6, \end{aligned} \]

  the area of the rectangle.

[1]

P. Billingsley, Probability and measure, 3rd ed. in Wiley series in probability and mathematical statistics. New York: Wiley, 1995.