A singular measure on the Cantor group ✩

Let Ω = {−1, 1}N and {ωj } be independent random variables taking values in {−1, 1} with equal probability. Endowed with the product topology and under the operation of pointwise product, Ω is a compact Abelian group, the so-called Cantor group. Let a, b, c be real numbers with 1 + a + b + c >0, 1+a − b −c >0, 1− a + b −c >0 and 1− a − b +c >0. Finite products on Ω, Pn = n j=1 (1+ aωj + bωj+1 +cωjωj+1), are studied.We show that the weak limit of Pn dω Ω Pn dω exists in the topology of M(Ω), where M(Ω) is the convolution algebra of all Radon measure on Ω, thus defined a probability measure on Ω. We also prove that the measure is continuous and singular with respect to the normalized Haar measure on Ω.  2005 Elsevier Inc. All rights reserved.