---
tag: GSFXAV0VD7
kind: concept
name: Universally measurable set
msc: [28A05, 28C15]
sources:
  - url: https://en.wikipedia.org/wiki/Universally_measurable_set
same_as:
  wikidata: Q7894192
  wikipedia: Universally measurable set
---

## Summary

A subset of a measurable space is universally measurable if it is measurable for the completion of every probability measure on the space.
The universally measurable sets form a σ-algebra that contains all the original measurable sets.
In Polish spaces they include the analytic sets, which may fail to be Borel.

## Explanations

### L1 human

A set may fail to be in the original σ-algebra and still be harmless for every measure one could put on the space.
Universally measurable sets are those sets: whatever probability measure one chooses, the set differs from a measurable set by a negligible piece.
Every Borel subset of the real line is universally measurable.
A Vitali set is not, because it is not measurable for Lebesgue measure on $[0, 1]$.