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