---
tag: KJNSKV75VQ
kind: concept
name: Completion of a measure
msc: [28A12]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_measure
---

## Summary

The completion of a measure space adds to the σ-algebra every set that differs from a measurable set by a subset of a null set.
The measure extends to these sets in only one way, and the result is the smallest complete extension.
Lebesgue measure on the Lebesgue measurable sets is the completion of Lebesgue measure on the Borel sets.

## Explanations

### L1 human

Some measures leave out subsets of negligible sets.
The completion puts them back, and gives each new set the measure of the measurable set it is almost equal to.
Nothing of substance changes: the new sets differ from old ones only by negligible pieces.
This is how one passes from Borel sets on the line to all Lebesgue measurable sets.