### Abstract

For r ∈ [0,1] let μ_{r} 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 μ_{r} is continuously reducible to μ_{s} (that is, there is a continuous map sending μ_{r} to μ_{s}) 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 μ_{r} and μ_{s} 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 language | English (US) |
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Pages (from-to) | 103-110 |

Number of pages | 8 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 142 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2007 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematical Proceedings of the Cambridge Philosophical Society*,

*142*(1), 103-110. https://doi.org/10.1017/S0305004106009741

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

Research output: Contribution to journal › Article

*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 142, no. 1, pp. 103-110. https://doi.org/10.1017/S0305004106009741

}

TY - JOUR

T1 - A pair of non-homeomorphic product measures on the Cantor set

AU - Austin, Tim D.

PY - 2007/1

Y1 - 2007/1

N2 - 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?

AB - 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?

UR - http://www.scopus.com/inward/record.url?scp=33846986475&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846986475&partnerID=8YFLogxK

U2 - 10.1017/S0305004106009741

DO - 10.1017/S0305004106009741

M3 - Article

VL - 142

SP - 103

EP - 110

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 1

ER -