---
tag: SV96KN4WR9
kind: concept
name: Non-atomic measure
msc: [28A12]
specializes: [PR11BZQQPG]
sources:
  - url: https://en.wikipedia.org/wiki/Atom_(measure_theory)
---

## Summary

An atom of a measure is a measurable set of positive measure that cannot be split into two parts of positive measure.
A measure is non-atomic, or atomless, if it has no atoms.
Non-atomic measures can be divided finely: by Sierpiński's theorem, a set of finite measure $m$ has subsets of every measure between $0$ and $m$.

## Explanations

### L1 human

An atom is a lump of mass that cannot be split.
A point mass, such as the Dirac measure at a point, is one big atom.
Length on the real line has no atoms: any set of positive length can be cut into two pieces of positive length.
A non-atomic measure is spread out like sand, not concentrated in lumps.