Let X be a complex Banach space, G a compact abelian group and Λ a subset of Ĝ, the dual group pf G. Then LΛ1(G, X) has the Radon-Nikodym property if and only if X has the Radon-Nikodym property and Λ is Riesz set. In particular, H1 (T, X) has the Radon-Nikodym property if and only if X has the Radon-Nikodym property. This solves a problem of Hensgen.