  Let $\Sigma$ be a $\sigma$-algebra over $\Omega$, and let $M(\Sigma)$ denote
the Banach space of complex measures. Consider a representation $T_t$ for
$t\in\Bbb R$ acting on $M(\Sigma)$. We show that under certain, very weak
hypotheses, that if for a given $\mu \in M(\Sigma)$ and all $A \in \Sigma$ the
map $t \mapsto T_t \mu(A)$ is in $H^\infty(\Bbb R)$, then it follows that the
map $t \mapsto T_t \mu$ is Bochner measurable. The proof is based upon the idea
of the Analytic Radon Nikod\'ym Property.
  Straightforward applications yield a new and simpler proof of Forelli's main
result concerning analytic measures ({\it Analytic and quasi-invariant
measures}, Acta Math., {\bf 118} (1967), 33--59).
