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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On smoothing compact measure spaces by multiplication

Author: Dorothy Maharam
Journal: Trans. Amer. Math. Soc. 204 (1975), 1-39
MSC: Primary 28A35; Secondary 28A60
MathSciNet review: 0374367
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Abstract: Let $ \mu $ be a regular Borel probability measure on a compact Hausdorff space $ S$, and let $ m$ be Lebesgue measure on the unit interval $ I$. It is proved that the measure-theoretic product $ (S,\mu ) \times ({I^w},{m^w})$, where $ w$ is a large enough cardinal and $ {m^w}$ denotes product Lebesgue measure, is ``pseudo-isometric'' to $ ({I^w},{m^w})$. Here a pseudo-isometry $ \phi $ is a point-isometry except that, instead of $ \phi (A)$ being measurable for every measurable $ A$, it is required only that $ A$ differ by a null set from a set with measurable image. If instead $ \mu $ is a Baire probability measure and $ S$ is a Baire subset of $ {I^w}$, then $ (S,\mu ) \times ({I^w},{m^w})$ is point-isometric to $ ({I^w},{m^w})$. Finally it is shown that (roughly speaking) continuous maps can be ``smoothed'' into projection maps (to within pseudo-isometries) by multiplication by suitable projection maps.

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Keywords: Regular Borel probability measure, Baire measures, Baire sets, Borel sets, point and set isometries of measure spaces, Baire and Borel isomorphisms, disintegration of measures, measure algebra, functions monotonic in one variable, evaluation map, measure-preserving transformations
Article copyright: © Copyright 1975 American Mathematical Society