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A pair of non-homeomorphic product measures on the Cantor set

Published online by Cambridge University Press:  12 February 2007

TIM D. AUSTIN
TIM D. AUSTIN*
Affiliation:
Department of Mathematics, UCLA, Los Angeles CA 90095-1555, U.S.A. e-mail: timaustin@math.ucla.edu
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Abstract

For r ∈ [0, 1] let μr be the Bernoulli measure on the Cantor set given as the infinite power of the measure on {0, 1} with weights r and 1 − r. For r, s ∈ [0, 1] it is known that the measure μr is continuously reducible to μs (that is, there is a continuous map sending μr to μs) if and only if s can be written as a certain kind of polynomial in r; in this case s is said to be binomially reducible to r. In this paper we answer in the negative the following question posed by Mauldin:

Is it true that the product measures μrand μsare homeomorphic if and only if each is a continuous image of the other, or, equivalently, each of the numbersrandsis binomially reducible to the other?

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 1 , January 2007 , pp. 103 - 110
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

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