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Denumerable compact metric spaces admit isometry-invariant finitely additive measures

Published online by Cambridge University Press:  26 February 2010

Roy O. Davies  and
A. J. Ostaszewski
Roy O. Davies
Affiliation:
Department of Mathematics, The University, Leicester, LE1 7RH.
A. J. Ostaszewski
Affiliation:
Department of Mathematics, London School of Economics, Houghton Street, London, WC2A 2AE.
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Extract

Let X be a compact metric space. By an invariant measure on X we shall mean a finitely additive non-negative real-valued function μ. on the Borel σ-algebra in X, such that μ(X) = 1 and Borel subsets of X have equal measure if there exists an isometry from one onto the other (not necessarily extendable to the whole of X). We note in passing that an isometric copy of a Borel set is necessarily itself Borel, by a well-known theorem of Souslin (see [3], §39. V). In Problem 2 of the Scottish Book (17. VII. 1935; see [2]), Banach and Ulam asked whether every non-empty compact metric space admits an invariant measure. This problem remains open; we give a positive answer in a veryspecial case.

Type
Research Article
Information
Mathematika , Volume 26 , Issue 2 , December 1979 , pp. 184 - 186
Copyright
Copyright © University College London 1979

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