TY - JOUR T1 - A Riesz Product Type Measure on the Cantor Group AU - Shi , Qiyan JO - Communications in Mathematical Research VL - 1 SP - 7 EP - 16 PY - 2021 DA - 2021/05 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19173.html KW - Riesz product, Cantor group, weak topology, singularity of measure. AB -

A Riesz type product as $$P_n = \prod\limits_{j=1}^n (1 + aω_j + bω_{j+1})$$ is studied, where $a, b$ are two real numbers with $|a| + |b| < 1$, and {$ω_j$} are independent random variables taking values in {−1, 1} with equal probability. Let d$ω$ be the normalized Haar measure on the Cantor group $Ω$ = {−1, 1}$^N$. The sequence of probability measures $\Big \{\frac{P_n{\rm d}ω}{E(P_n)} \Big \}$ is showed to converge weakly to a unique continuous measure on $Ω$, and the obtained measure is singular with respect to d$ω$.