Let \((S,+)\) be a locally compact Abelian semigroup, let \(\mu\) be a Borel probability measure on \(S\), and let \(T\) be the corresponding convolution operator defined, for every bounded Borel function \(f\) on \(S\), by \(Tf(s)=\int_ Sf(s+u)\mu(du)\) for all \(s\in S\). The author gives an elementary and elegantly brief proof of the following theorem of Choquet and Deny: A continuous (resp. Borel) function \(h\) satisfies \(Th=h\) if and only if we have \(h(s+u)=h(s)\) for all \(s\in S\), and for all \(u\) in the support of \(\mu\) (resp. \(\mu\)-almost all \(u)\).