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Bands of invariantly extensible measures. (English) Zbl 0692.28003

Author’s summary: “Given two \(\sigma\)-algebras \({\mathcal A}\subset {\mathcal B}\), invariant under a fixed semigroup G of transformations, the following subset C of the lattice cone M(\({\mathcal A})_ G\) of G-invariant finite measures on \({\mathcal A}\) is shown to be (the positive part of) a band in M(\({\mathcal A})_ G:\) A G-invariant measure \(\mu\) belongs to C iff the set \(exM({\mathcal B};\mu)_ G\) of extremal G-invariant extensions of \(\mu\) to \({\mathcal B}\) is non-empty and each G-invariant extension \(\nu\) of \(\mu\) admits a barycentric decomposition \(\nu =\int \nu '\rho (d\nu ')\) with some representing probability \(\rho\) on \(exM({\mathcal B};\mu)_ G.\)- Any band of extensible measures allows to study the corresponding extension problem locally.”
Reviewer: D.Plachky

MSC:

28A60 Measures on Boolean rings, measure algebras
28D05 Measure-preserving transformations
28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
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References:

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