---
tag: 6HVCTJGB61
kind: concept
name: Product sigma-algebra
msc: [28A05, 28A35]
specializes: [0SV8K0TDGY]
sources:
  - url: https://en.wikipedia.org/wiki/%CE%A3-algebra
---

## Summary

The product σ-algebra on a Cartesian product of [[measurable spaces|N4KDT83V3S]] is the smallest [[σ-algebra|0SV8K0TDGY]] that makes every coordinate projection measurable.
For two factors it is generated by the measurable rectangles $A \times B$.
It is the σ-algebra on which [[product measures|77PV0ZAVXT]] live and on which Fubini's theorem is stated.
For infinite products it is generated by the cylinder sets, which constrain only finitely many coordinates.

## Explanations

### L1 human

To measure sets in a product $X \times Y$, start from the rectangles $A \times B$ built from measurable pieces of each factor.
The product σ-algebra contains these rectangles and everything one can build from them with countable unions, countable intersections and complements.
For the real line with its Borel sets, this produces exactly the Borel sets of the plane.
For infinitely many coin tosses, it contains the events that depend on finitely many tosses, and countable combinations of them, such as "heads appears infinitely often".