---
tag: PKQ7ABPVSG
kind: concept
name: Non-measurable set
msc: [28A05]
sources:
  - url: https://en.wikipedia.org/wiki/Non-measurable_set
same_as:
  wikidata: Q1055379
  wikipedia: Non-measurable set
---

## Summary

A non-measurable set is a set to which a given measure cannot assign a size, because it lies outside the $\sigma$-algebra of measurable sets.
The term most often refers to subsets of $\R^n$ that are not measurable for [[Lebesgue measure|2Z2QVWZ262]].
Their existence shows that length, area and volume cannot be defined for all sets in a way that is countably additive and invariant under translations.
Their construction needs some form of the [[axiom of choice|0HTGB13FE2]].
Solovay showed that ZF with only dependent choice cannot prove that they exist, if an inaccessible cardinal is consistent with ZFC.

## Explanations

### L1 human

Length works well for intervals and for all sets one usually meets.
But with the axiom of choice one can build sets of real numbers so scattered that no length fits them.
The classical example is a [[Vitali set|WYHGXY471C]], which picks one number from each class of numbers that differ by a fraction.
Shifted copies of it would have to have equal lengths that add up to a finite positive number, which no single length can do.