Math 2.0

Measure and integration Definition

Radon measure

Builds on Hausdorff spaceBorel measureLocally finite measureInner regular measureCompact set

Definition

Let XX be a Hausdorff space and let μ\mu be a Borel measure on XX. The measure μ\mu is a Radon measure if both of the following hold.

  1. μ\mu is locally finite: every point has an open neighbourhood of finite measure.
  2. μ\mu is inner regular: μ(A)=sup⁡{μ(K):K⊆A, K compact}\mu(A) = \sup \set{\mu(K) : K \subseteq A, \ K \text{ compact}} for every Borel set AA.

Remarks

Example: the Dirac measure δp\delta_p at a point pp is a Radon measure. Every point has a neighbourhood of measure at most 11; and if p∈Ap \in A then K={p}K = \{p\} is a compact subset of AA with δp(K)=δp(A)\delta_p(K) = \delta_p(A), while if p∉Ap \notin A both sides are 00. Lebesgue measure on ℝn\R^n is also a Radon measure.

Non-example: counting measure on ℝ\R satisfies clause 2 but not clause 1.

Another common definition, used on locally compact Hausdorff spaces, asks that μ\mu be finite on compact sets, outer regular on all Borel sets and inner regular on open sets. On a locally compact Hausdorff space in which every open set is σ-compact, such as ℝn\R^n, every Borel measure that is finite on compact sets is a Radon measure.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Borel measure, concept, no evidence levelBorel measureconceptCompact set, concept, no evidence levelCompact setconceptHausdorff space, concept, no evidence levelHausdorff spaceconceptInner regular measure, concept, no evidence levelInner regular measureconceptLocally finite measure, concept, no evidence levelLocally finite measureconceptRadon measure, definition, no evidence levelRadon measuredefinition
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
  1. Borel measure concept
  2. Compact set concept
  3. Hausdorff space concept
  4. Inner regular measure concept
  5. Locally finite measure concept
  6. Radon measure definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Radon_measure
About this page
TagFXN98YEQRZ
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Kinddefinition
WritingW0 stub
MSC28C15, 28C05
DefinesRadon measure
UsesHausdorff space, Borel measure, Locally finite measure, Inner regular measure, Compact set
Changed2 November 2026

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