Analytic measures and Bochner measurability

Abstract

Let $Σ$ be a $σ$-algebra over $Ω$, and let $M(Σ)$ denote the Banach space of complex measures. Consider a representation $T_t$ for $t\in\Bbb R$ acting on $M(Σ)$. We show that under certain, very weak hypotheses, that if for a given $μ\in M(Σ)$ and all $A \in Σ$ the map $t \mapsto T_t μ(A)$ is in $H^\infty(\Bbb R)$, then it follows that the map $t \mapsto T_t μ$ 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).