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

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?