Analytic measures and Bochner measurability

Abstract

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 $t\in \mathbb{R}$ 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 $H^{\infty }(\mathbb{R})$, 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 Nikodým 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).