---
tag: 4W9NM803BB
kind: concept
name: Pi-system
msc: [28A05, 60A10]
sources:
  - url: https://en.wikipedia.org/wiki/Pi-system
same_as:
  wikidata: Q3962249
  wikipedia: Pi-system
---

## Summary

A π-system on a set is a nonempty family of subsets that is closed under intersections of two members.
It is the kind of generating family used in the uniqueness theorems of measure theory.
Two probability measures that agree on a π-system agree on the σ-algebra it generates.

## Explanations

### L1 human

A π-system is a family of sets in which the overlap of any two members is again a member.
The rays $(-\infty, a]$ on the real line are the key example: the overlap of two rays is the shorter one.
These rays generate all Borel sets.
So a probability distribution on the line is fixed by its values on the rays, which is why a distribution function determines the distribution.