Saeki’s result states that on any locally compact nondiscrete group there exist continuous singular measures, with respect to the left Haar measure,
μ
\mu
with
μ
∗
μ
\mu * \mu
in
L
p
L^{p}
for all
p
,
1
≤
p
>
∞
p, \;1\leq p > \infty
. This paper gives a new and short proof of this using Rademacher-Riesz products.