Transference for amenable actions

Suppose $G$ acts amenably on a measure space $X$ with quasi-invariant $\sigma$-finite measure $m$. Let $\sigma$ be an isometric representation of $G$ on $L^p(X,dm)$ and $\mu$ a finite Radon measure on $G$. We show that the operator $\sigma_\mu f(x)=\int_G(\sigma(g)f)(x)d\mu(g)$ has $L^p(X,dm)$-operator norm not exceeding the $L^p(G)$-operator norm of the convolution operator defined by $\mu$. We shall also prove an analogous result for the maximal function $M$ associated to a countable family of Radon measures $\mu_n$.