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

Tim D. Austin

Research output: Contribution to journalArticle

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 μr and μs are homeomorphic if and only if each is a continuous image of the other, or, equivalently, each of the numbers r and s is binomially reducible to the other?

Original languageEnglish (US)
Pages (from-to)103-110
Number of pages8
JournalMathematical Proceedings of the Cambridge Philosophical Society
Volume142
Issue number1
DOIs
StatePublished - Jan 2007

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Product Measure
Cantor set
If and only if
Continuous Image
Continuous Map
Homeomorphic
Bernoulli
Polynomial

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

A pair of non-homeomorphic product measures on the Cantor set. / Austin, Tim D.

In: Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 142, No. 1, 01.2007, p. 103-110.

Research output: Contribution to journalArticle

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