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Erschienen in:
2019 | OriginalPaper | Buchkapitel
22. Measure Preserving Maps on Compact Hausdorff Spaces
verfasst von : Vittorino Pata
Erschienen in: Fixed Point Theorems and Applications
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Abstract
Let X be a compact Hausdorff space, and let P(X) be the set of all Borel probability measures on X. By means of the Riesz representation theorem, the dual space of \({\mathcal C}(X)\) can be identified with the space M(X) of complex regular Borel measures on X. Recall that the norm \(\Vert \mu \Vert \) of an element \(\mu \in M(X)\) is given by the total variation of \(\mu \). It is straightforward to check that P(X) is convex and closed in the weak\(^*\) topology of M(X).