  Let $\S$ be a commutative semigroup with identity $e$ and let $\Gamma$ be a
compact subset in the pointwise convergence topology of the space $\S'$ of all
non-zero multiplicative functions on $\S.$ Given a continuous function $F:
\Gamma \to \mathbb C$ and a complex regular Borel measure $\mu$ on $\Gamma$
such that $\mu(\Gamma) \not = 0.$ It is shown that $$ \mu(\Gamma) \int_{\Gamma}
\varrho(s) \overline{\varrho(t)} |F|^2(\varrho) d\mu(\varrho) = \int_{\Gamma}
\varrho(s) F(\varrho) d\mu(\varrho) \int_{\Gamma} \overline{\varrho(t)
F(\varrho)} d\mu(\varrho) $$ for all $(s, t) \in \S\times \S$ if and only if
for some $\gamma \in \Gamma, $ the support of
  $\mu$ is contained is contained in $\{ F = 0 \} \cup \{\gamma\}$. Several
applications of this characterization are derived. In particular, the reduction
of our theorem to the semigroup of non-negative integers $(\mathbb N_{0}, +)$
solves a problem posed by El Fallah, Klaja, Kellay, Mashregui and Ransford in a
more general context.
  More consequences of our results are given, some of them illustrate the
probabilistic flavor behind the problem studied herein and others establish
extremal properties of analytic kernels.
