---
tag: 3T6WG315GC
kind: concept
name: Metric outer measure
msc: [28A12, 28A78]
specializes: [BJ314H17HC]
sources:
  - url: https://en.wikipedia.org/wiki/Metric_outer_measure
same_as:
  wikidata: Q6824326
  wikipedia: Metric outer measure
---

## Summary

A metric outer measure is an [[outer measure|BJ314H17HC]] on a [[metric space|0VKD1TP7F3]] with one extra property.
It is additive on any two sets lying at positive distance from each other.
This condition links the outer measure with the geometry of the space.
Its main use is that for a metric outer measure every Borel set is measurable in the sense of Carathéodory.
Lebesgue outer measure and the [[Hausdorff outer measures|Q1TGZK8RN6]] are metric outer measures.

## Explanations

### L1 human

If two sets are separated by a gap, their total size should be the sum of their sizes.
A metric outer measure is an outer measure that respects this rule.
For example, in the plane, two disks that do not touch have total area equal to the sum of their areas.
This simple rule is enough to make all open and closed sets measurable.