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outer regular
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(Definition)
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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 $$ \mu(B) = \inf\,\{ \mu(U) \mid U \supset B,\ U\ \rm{open} \}. $$ We say $\mu$
is inner regular on $B$ if $$ \mu(B) = \sup\,\{ \mu(K) \mid K \subset B,\ K\ \rm{compact} \}. $$
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"outer regular" is owned by djao.
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Cross-references: Borel set, measure, Hausdorff topological space, locally compact
There are 4 references to this entry.
This is version 1 of outer regular, born on 2002-05-26.
Object id is 2938, canonical name is OuterRegular.
Accessed 4758 times total.
Classification:
| AMS MSC: | 28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities) |
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Pending Errata and Addenda
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