Smoothness conditions on measures using Wallman spaces

Charles Traina

International Journal of Mathematics and Mathematical Sciences, 1999, vol. 22, 1-14

Abstract:

In this paper, X denotes an arbitrary nonempty set, ℒ a lattice of subsets of X with ∅ , X ∈ ℒ , A ( ℒ ) is the algebra generated by ℒ and M ( ℒ ) is the set of nontrivial, finite, and finitely additive measures on A ( ℒ ) , and M R ( ℒ ) is the set of elements of M ( ℒ ) which are ℒ -regular. It is well known that any μ ∈ M ( ℒ ) induces a finitely additive measure μ ¯ on an associated Wallman space. Whenever μ ∈ M R ( ℒ ) , μ ¯ is countably additive.

We consider the general problem of given μ ∈ M R ( ℒ ) , how do properties of μ ¯ imply smoothness properties of μ ? For instance, what conditions on μ ¯ are necessary and sufficient for μ to be σ -smooth on ℒ , or strongly σ -smooth on ℒ , or countably additive? We consider in discussing these questions either of two associated Wallman spaces.

Date: 1999
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:734302

DOI: 10.1155/S0161171299227135

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