---
tag: 77PV0ZAVXT
kind: concept
name: Product measure
msc: [28A35]
sources:
  - url: https://en.wikipedia.org/wiki/Product_measure
same_as:
  wikidata: Q1572094
  wikipedia: Product measure
---

## Summary

The product measure $\mu \times \nu$ of two [[σ-finite|WEBBT18D4X]] measures is the measure on the [[product σ-algebra|6HVCTJGB61]] that gives each rectangle $A \times B$ the measure $\mu(A) \nu(B)$.
It exists and is unique under σ-finiteness.
It is the setting of Fubini's and Tonelli's theorems, which compute integrals over the product as iterated integrals.
In probability it describes pairs of independent random variables.

## Explanations

### L1 human

The area of a rectangle is width times height.
The product measure extends this rule from rectangles to all reasonable subsets of a product space.
Lebesgue measure on the plane is the product of Lebesgue measure on the line with itself.
If one throws a fair coin and a fair die independently, the probability of "heads and a six" is $\tfrac12 \cdot \tfrac16 = \tfrac{1}{12}$, a value of the product measure.