---
tag: M8X6VZD91K
kind: definition
name: Product sigma-algebra
msc: [28A05, 28A35]
defines: 6HVCTJGB61
uses: [0SV8K0TDGY, N4KDT83V3S, 09CEDJYBKP, YYNCJSR0PJ]
sources:
  - url: https://en.wikipedia.org/wiki/%CE%A3-algebra
---

## Definition

Let $I$ be a nonempty index set and, for each $i \in I$, let $(X_i, \Sigma_i)$ be a [[measurable space|N4KDT83V3S]].
Let $X = \prod_{i \in I} X_i$ be the [[Cartesian product|09CEDJYBKP]], and let $\pi_i \colon X \to X_i$ be the coordinate projections.
The *product σ-algebra* $\bigotimes_{i \in I} \Sigma_i$ is the [[σ-algebra|0SV8K0TDGY]] on $X$ generated by the sets $\pi_i^{-1}(A)$ with $i \in I$ and $A \in \Sigma_i$.
Equivalently, it is the smallest σ-algebra on $X$ for which every $\pi_i$ is [[measurable|YYNCJSR0PJ]].

## Remarks

For two factors, $\Sigma_1 \otimes \Sigma_2$ is generated by the rectangles $A_1 \times A_2$ with $A_1 \in \Sigma_1$ and $A_2 \in \Sigma_2$, since $A_1 \times A_2 = \pi_1^{-1}(A_1) \cap \pi_2^{-1}(A_2)$.
For general $I$, it is generated by the cylinder sets $\prod_{i \in J} A_i \times \prod_{i \notin J} X_i$ with $J \subseteq I$ finite.

Example: $\mathcal{B}(\R) \otimes \mathcal{B}(\R) = \mathcal{B}(\R^2)$, where $\mathcal{B}$ denotes the [[Borel σ-algebra|0WQXY32Q8D]].
Each rectangle of Borel sets is Borel in $\R^2$, which gives one inclusion.
Each open set of $\R^2$ is a countable union of open rectangles with rational corners, which gives the other.

Non-example: let $\mathcal{L}$ be the σ-algebra of Lebesgue measurable subsets of $\R$.
Then $\mathcal{L} \otimes \mathcal{L}$ is strictly smaller than the σ-algebra of Lebesgue measurable subsets of $\R^2$.
For a non-measurable $N \subseteq \R$, the set $\set{0} \times N$ is a Lebesgue null set in $\R^2$, but its section at $0$ is $N \notin \mathcal{L}$, and every set in $\mathcal{L} \otimes \mathcal{L}$ has measurable sections.

A function $f \colon Z \to X$ is measurable for $\bigotimes_i \Sigma_i$ if and only if every coordinate $\pi_i \circ f$ is measurable.