---
tag: PR11BZQQPG
kind: concept
name: Measure
msc: [28A12, 28A10]
sources:
  - url: https://en.wikipedia.org/wiki/Measure_(mathematics)
same_as:
  wikidata: Q192276
  wikipedia: Measure (mathematics)
---

## Summary

A measure on a measurable space is a function that gives each measurable set a size in $[0, \infty]$.
It gives the empty set size $0$ and is countably additive: the size of a countable disjoint union is the sum of the sizes.
Measures generalize length, area, volume, counting and probability, and they are the basis of the Lebesgue integral.

## Explanations

### L1 human

A measure is a consistent way to give sizes to sets.
If a set is cut into countably many separate pieces, its size is the sum of the sizes of the pieces.
Length on the real line is a measure: the interval $[0, 2]$ has size $2$, the union of $[0, 1)$ and $[1, 2]$.
Counting the elements of a set is also a measure, and so is probability.
Sizes may be infinite, as for the length of the whole line.