---
tag: EBGECZNCV5
kind: concept
name: Mutually singular measures
msc: [28A10]
sources:
  - url: https://en.wikipedia.org/wiki/Singular_measure
same_as:
  wikidata: Q1824450
  wikipedia: Singular measure
---

## Summary

Two [[measures|PR11BZQQPG]] on the same space are mutually singular when they live on disjoint parts of the space.
One of them gives zero mass to a set, and the other gives zero mass to its complement.
Mutual singularity is the opposite extreme of [[absolute continuity of measures|P0P21KDV6E]].
The [[Lebesgue decomposition theorem|74VMD07DRX]] splits a σ-finite measure, relative to a σ-finite reference measure, into an absolutely continuous part and a singular part.

## Explanations

### L1 human

Picture two piles of sand on the same floor that never touch.
Each pile sits on its own region, and the two regions do not overlap.
On the real line, Lebesgue measure spreads mass evenly, while the point mass at $0$ puts all its mass on the single point $0$.
Lebesgue measure gives the set $\{0\}$ length zero, and the point mass gives the rest of the line mass zero.
So these two measures are mutually singular.