A pair of non-homeomorphic product measures on the Cantor set
Abstract
For r in [0,1] let μr be the Bernoulli measure on the Cantor set given as the infinite power of the measure on the two-point set with weights r and 1-r. For r and s in [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?
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