A singular measure on the Cantor group

Let Ω=N{−1,1} 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,