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outer regular (Definition)

Let $ X$ be a locally compact Hausdorff topological space with Borel $ \sigma$-algebra $ \mathcal{B}$, and suppose $ \mu$ is a measure on $ (X,\mathcal{B})$. For any Borel set $ B \in \mathcal{B}$, the measure $ \mu$ is said to be outer regular on $ B$ if

$\displaystyle \mu(B) = \inf\,\{ \mu(U) \mid U \supset B,\ U\ \rm {open} \}. $
We say $ \mu$ is inner regular on $ B$ if
$\displaystyle \mu(B) = \sup\,\{ \mu(K) \mid K \subset B,\ K\ \rm {compact} \}. $



"outer regular" is owned by djao.
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See Also: Borel $\sigma$-algebra

Also defines:  inner regular
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Cross-references: Borel set, measure, Hausdorff topological space, locally compact
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This is version 1 of outer regular, born on 2002-05-26.
Object id is 2938, canonical name is OuterRegular.
Accessed 3873 times total.

Classification:
AMS MSC: 28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities)
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