# Baire measure

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## Baire measure

[′ber ‚mezh·ər]
(mathematics)
A measure defined on the class of all Baire sets such that the measure of any closed, compact set is finite.
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In this case we prove that any Baire measure can be uniquely extended to regular Borel measure.
Suppose X is a Hausdorff normal and countably paracompact topological space and [mu]: [B.sub.0](X) [right arrow] E a Baire measure. 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.
Since [mu] is a Baire measure, we have 0 = [[bar.[mu]].sub.p]([intersection]([Z.sub.n]).
695, there is v: [B.sub.0](X) [right arrow] E, a unique Baire measure, satisfying all the conditions of the above theorem, except Borel extension.
The elements of the [sigma]-algebra generated by zero-sets are called Baire sets and the elements of the [sigma]-algebra generated by closed sets are called Borel sets; B(X)and [B.sub.0](X) are the classes of Borel and Baire subsets of X and [M.sub.[sigma]](X) denotes the class of all scalar-valued, countably additve Baire measures on X.
Wheeler, "A survey of Baire measures and strict topologies," Expositiones Mathematicae, vol.
Among the results obtained in the domain of Vector Measures and Vector Integration, we quote the already mentioned facts and (the selection is clearly subjective): Vector Radon-Nikodym-type theorems, integral representations of linear operators on function spaces, results concerning the Dunford and Pettis integrals, results concerning the regularity of vector Baire measures and results concerning compactness in spaces of measures.

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