Structural properties of compact groups with measure-theoretic applications. (English) Zbl 0831.28007
The present paper establishes a close connection between compact groups and products of compact metric spaces. Let \(G\) denote a compact topological group. The author proves that there exists a family \((\mu_\lambda)_{\lambda> \alpha}\) of Radon probability measures each supported on a compact metric space \(X_\lambda\) having at least two points and a Baire isomorphism from the product \(\prod_{\lambda< \alpha} X_\lambda\) onto \(G\) whose induced isomorphism of measures takes the product measure \(\otimes(\mu_\lambda)_{\lambda< \alpha}\) to normalized Haar measure on \(G\).
Throughout the paper \(\alpha\), the topological weight of \(G\), is assumed uncountable. In the last section of the paper two applications of the theorem are presented.
Throughout the paper \(\alpha\), the topological weight of \(G\), is assumed uncountable. In the last section of the paper two applications of the theorem are presented.
Reviewer: B.B.Wells jun.(McLean)
MSC:
| 28C10 | Set functions and measures on topological groups or semigroups, Haar measures, invariant measures |
| 43A10 | Measure algebras on groups, semigroups, etc. |
| 43A05 | Measures on groups and semigroups, etc. |
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