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Analytic measures and Bochner measurability

N. Asmar·Stephen J. Montgomery-Smith
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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).

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N. Asmar, Stephen J. Montgomery-Smith. 1999-12-06. Analytic measures and Bochner measurability. https://arxiv.org/abs/math/9501211

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