---
tag: P15VQS5DJ9
kind: concept
name: Measure space
msc: [28A12]
sources:
  - url: https://en.wikipedia.org/wiki/Measure_space
same_as:
  wikidata: Q3058212
  wikipedia: Measure space
---

## Summary

A measure space is a set together with a σ-algebra of subsets and a measure on that σ-algebra.
It is the basic object of measure theory and of the Lebesgue integral.
A probability space is a measure space of total measure $1$.

## Explanations

### L1 human

A measure space has three parts: a set of points, a list of subsets that one may measure, and a rule that gives each of them a size.
The real line with the Borel sets and length is the standard picture.
A fair die gives a small example: the six faces, all subsets of faces, and the size of a subset is its number of faces divided by $6$.