---
tag: 28E5Q5BM69
kind: concept
name: Inner regular measure
msc: [28C15]
specializes: [DYR1QFNGNJ]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_measure
---

## Summary

A measure on a topological space is inner regular when each measurable set is approximated from inside by compact subsets.
The approximation is in measure.
Inner regularity lets one reduce questions about general sets to questions about compact sets, where topology gives control.
It is one of the two halves of the definition of a Radon measure.
Some authors use closed sets instead of compact sets in the definition.

## Explanations

### L1 human

To measure a complicated set, fill it from inside with compact pieces, like filling a lake with larger and larger islands.
If the islands can capture as much of the measure as one likes, the measure is inner regular.
For length on the real line, the set of irrational numbers in $[0, 1]$ has length $1$.
It contains compact sets, built by removing small open intervals around the rationals, of length as close to $1$ as one likes.