Koopman representations for positive definite functions

We show that for any locally compact second countable group $G$ and any continuous positive definite function $\phi :G\to \mathbb{C}$, there exists an ergodic measure preserving system $(X,\mathcal{B},\mu ,\{T_g{\}}_{g\in G})$ and a function $f\in L^2(X,\mu )$ for which $\phi (g)=⟨T_{g}f,f⟩$. We also show that if $G$ is a countably infinite abelian group, then there exists a (not necessarily ergodic) measure preserving system $(X,\mathcal{B},\mu ,\{T_g{\}}_{g\in G})$ and a function $f\in L^2(X,\mu )$ with $\mathrm{\mid }f\mathrm{\mid }=\phi (0)$ and $\phi (g)=⟨T_{g}f,f⟩$.