Regular measure

Builds on Borel measure Inner regular measure Outer regular measure

Definition

Let $X$ be a topological space and let $\mu$ be a Borel measure on $X$ . The measure $\mu$ is regular if both of the following hold.

$\mu$ is inner regular: $\mu(A) = \sup \set{\mu(K) : K \subseteq A, \ K \text{ compact and Borel}}$ for every Borel set $A$ .
$\mu$ is outer regular: $\mu(A) = \inf \set{\mu(U) : U \supseteq A, \ U \text{ open}}$ for every Borel set $A$ .

Remarks

Example: Lebesgue measure on the Borel sets of $\R^n$ is regular. Outer regularity follows from its construction by covers with open boxes. For inner regularity, a Borel set $A$ contains a closed set $F$ with $\lambda(A \setminus F)$ arbitrarily small when $\lambda(A) < \infty$ . The sets $F \cap [-k, k]^n$ are compact, and their measures tend to $\lambda(F)$ . The case $\lambda(A) = \infty$ follows by cutting $A$ into bounded pieces.

Non-example: counting measure on $\R$ satisfies clause 1 but not clause 2. Finite sets are compact, which gives clause 1; the set $\{0\}$ has measure $1$ , but every open set around it has measure $\infty$ .

Conventions vary. Many texts use closed sets instead of compact sets in clause 1. On a metric space, every finite Borel measure is regular in that closed-set sense.

Reading path

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Tag	`3EC1CQKMK2`
URI	`m2://3EC1CQKMK2`
Kind	definition
Writing	W0 stub
MSC	28C15
Defines	Regular measure
Uses	Borel measure, Inner regular measure, Outer regular measure
Changed	2 November 2026

Regular measure

Definition

Remarks

Reading path

Further reading

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