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=∏j=1n(1+aωj+bωj+1+cωjωj+1), are studied. We show that the weak limit of {Pndω∫ΩPndω} 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.