Nakhlé Asmar and Stephen Montgomery-Smith, Analytic measures and Bochner measurability.
Bull. Sc. Math., 122, (1998), 39-66.
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 Nikodym Property. Straightforward applications yield a new and simpler proof of Forelli's main result concerning analytic measures (Analytic and quasi-invariant measures, Acta Math., 118 (1967), 33--59).
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