---
tag: 9ZP4ER2N65
kind: definition
name: Product measure
msc: [28A35]
defines: 77PV0ZAVXT
uses: [PR11BZQQPG, WEBBT18D4X, 6HVCTJGB61]
sources:
  - url: https://en.wikipedia.org/wiki/Product_measure
---

## Definition

Let $(X, \Sigma, \mu)$ and $(Y, T, \nu)$ be measure spaces, and assume that $\mu$ and $\nu$ are [[σ-finite|WEBBT18D4X]].
Let $\Sigma \otimes T$ be the [[product σ-algebra|6HVCTJGB61]] on $X \times Y$.
The *product measure* $\mu \times \nu$ is the unique [[measure|PR11BZQQPG]] on $\Sigma \otimes T$ with
$$(\mu \times \nu)(A \times B) = \mu(A) \, \nu(B) \quad \text{for all } A \in \Sigma, \ B \in T,$$
with the convention $0 \cdot \infty = 0$.

## Remarks

Existence follows from [[Carathéodory's extension theorem|Q8EXG7EKYW]] applied to the rectangles, and uniqueness from σ-finiteness and the π-λ theorem.
For $E \in \Sigma \otimes T$ one has $(\mu \times \nu)(E) = \int_X \nu(E_x) \, d\mu(x) = \int_Y \mu(E^y) \, d\nu(y)$, where $E_x$ and $E^y$ are the sections of $E$.

Example: let $\mu$ and $\nu$ be Lebesgue measure on the Borel sets of $[0, 1]$.
Then $\mu \times \nu$ is two-dimensional Lebesgue measure on the Borel sets of $[0, 1]^2$: both measures give each rectangle its area, so they agree by uniqueness.

Non-example: let $\delta$ be the image of Lebesgue measure $\lambda$ on $[0, 1]$ under $x \mapsto (x, x)$, a measure on the diagonal of $[0, 1]^2$.
For $A = [0, \tfrac12]$ and $B = [\tfrac12, 1]$, $\delta(A \times B) = \lambda(A \cap B) = 0$, but $\lambda(A) \lambda(B) = \tfrac14$.
So $\delta$ is not $\lambda \times \lambda$.

Without σ-finiteness, uniqueness can fail.
For Lebesgue measure on $[0, 1]$ and counting measure on $[0, 1]$, there are different measures on the product σ-algebra with the rectangle property.
The product of Lebesgue measures is often completed to give Lebesgue measure on $\R^{m+n}$.