Inner regular measure

Definition

Let $X$ be a topological space and let $\mu$ be a Borel measure on $X$ . A Borel set $A \subseteq X$ is inner regular for $\mu$ if $\mu(A) = \sup \set{\mu(K) : K \subseteq A, \ K \text{ compact and Borel}}.$ The measure $\mu$ is inner regular if every Borel set is inner regular for $\mu$ .

Remarks

In a Hausdorff space every compact set is closed, hence Borel, so the condition "Borel" on $K$ is automatic.

Example: counting measure on $\R$ is inner regular. If $A$ is finite, take $K = A$ ; if $A$ is infinite, its finite subsets are compact and have arbitrarily large counting measure, so the supremum is $\infty = \mu(A)$ .

Non-example: let $X$ be an uncountable set with the cocountable topology, whose open sets are $\emptyset$ and the complements of countable sets. Its Borel sets are the countable sets and their complements; let $\mu$ be $0$ on countable sets and $1$ on the others. Compact subsets of $X$ are finite, so they have measure $0$ , while $\mu(X) = 1$ ; thus $\mu$ is not inner regular.

Variants: some authors approximate by closed sets instead of compact sets, and some ask inner regularity only for open sets.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Borel measure concept
Compact set concept
Inner regular measure definition

Tag	`SSFYQK86PK`
URI	`m2://SSFYQK86PK`
Kind	definition
Writing	W0 stub
MSC	28C15
Defines	Inner regular measure
Uses	Borel measure, Compact set
Changed	2 November 2026

Inner regular measure

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here