---
tag: YZA1V04R96
kind: concept
name: Measure algebra
msc: [28A60, "06E10"]
specializes: [0XADJKM699]
sources:
  - url: https://en.wikipedia.org/wiki/Measure_algebra
same_as:
  wikidata: Q6804179
  wikipedia: Measure algebra
---

## Summary

A measure algebra is a [[Boolean algebra|0XADJKM699]] in which countable suprema exist, together with a countably additive [[measure|PR11BZQQPG]] that vanishes only at $0$.
The main example is the algebra of measurable sets of a [[measure space|P15VQS5DJ9]] modulo [[null sets|VQQM4Y550T]].
Passing to the measure algebra forgets the points and keeps only what the measure can see.
By the [[isomorphism theorem for measure algebras|YFAQGE2ATQ]], all separable atomless probability measure algebras are isomorphic to that of $[0, 1]$.

## Explanations

### L1 human

In measure theory, two sets that differ only by a set of measure zero usually behave the same.
A measure algebra glues such sets together, so that each element is a measurable set "up to measure zero".
For instance, the intervals $[0, 1]$ and $(0, 1)$ become the same element of the measure algebra of Lebesgue measure.
What is left is an algebraic object on which the measure is still defined and is positive on every nonzero element.