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 - <p style="text-align: justify;">A Riesz type product as $$P_n =&nbsp;\prod\limits_{j=1}^n (1 + aω_j + bω_{j+1})$$ is studied, where $a, b$ are two real numbers with $|a| + |b| &lt; 1$, 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 $\Big \{\frac{P_n{\rm d}ω}{E(P_n)} \Big \}$&nbsp;is showed to converge weakly to a unique continuous
measure on $Ω$, and the obtained measure is singular with respect to d$ω$.</p>