DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Radon measure on an LCH space

Definition

For this page, a Borel measure $μ$ on a locally compact Hausdorff space is Radon when $μ (K) < \infty$ for every compact $K$ , and, for every Borel $E$ and every open $U$ , $μ (E) = \inf_{E \subseteq V open} μ (V), μ (U) = \sup_{K \subseteq U compact} μ (K) .$ Cohn calls this convention regular. It does not assert compact inner approximation for arbitrary Borel $E$ .

Depends on

Used by

Regular Borel measure on an LCH space Definition
Every Borel measure on an LCH space is Radon False statement
Inner regularity on open sets implies inner regularity on all Borel sets False statement
Lebesgue--Stieltjes regularity agrees with the LCH Radon convention on R Proposition
C_c(X) is dense in Lᵖ(mu) for a Radon measure Theorem
Lebesgue measure is a Radon measure on Rⁿ Theorem
Lusin's theorem for a Radon measure Theorem
Sigma-compact open sets make locally finite Borel measures regular Theorem
The RMK representing measure is inner regular on open sets Theorem
Uniqueness of the RMK representing measure among Radon measures Theorem

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Donald L. Cohn, Measure Theory, 2nd ed., §7.2 (standard reference, not scraped)