Radon measure

(redirected from Outer regular)

Radon measure

[′rā‚dän ‚mezh·ər]
(mathematics)
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i) v is inner regular by closed sets and outer regular by open sets;
A), A being any element in A with B = A [intersection] X; it is a trivial verification that v is well-defined, is finitely additive and it is inner regular by closed sets in X and outer regular by open sets in X (this means for a p [member of] P, F [member of] F and c > 0, there is, in X, a closed set C and an open set V, C [subset] F and V [contains] F, such that for any [B.
Let v: F [right arrow] E be a finitely additive regular (inner regular by closed sets in X and outer regular by open sets in X) measure, having relatively weakly compact range, such that [integral] fdv = 0, [for all]f [member of] [C.
Then it has a unique extension to a countably additive Borel measure [mu]: B(X) [right arrow] E which is inner regular by closed sets and outer regular by open sets.
intersection] X = B; define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; it is a trivial verification that [mu] is well-defined, is countably additive and it is inner regular by closed sets in X and outer regular by open sets in X.