On the mean value of a weakly almost periodic function

Abstract:

Let M denote the invariant mean on the space $W(G)$ of weakly almost periodic functions on a LCA group G. The purpose of this note is to show that, for each $\phi \in W(G)$, \begin{equation}\tag {$1$}M(\phi ) = \lim \limits _{V \to \{ 1\} } \int _G {{{\hat f}_V}(x)\phi (x)dx} \end{equation} where {V} is the system of compact neighborhoods of 1 in the character group $\Gamma$, and, for each V, ${f_V}$ is a continuous positive definite function supported in V and satisfying ${f_V}(1) = 1$. This answers affirmatively a question recently raised by R. Burckel.