On the near differentiability property of Banach spaces

Abstract

Let μ be a scalar measure of bounded variation on a compact metrizable abelian group G. Suppose that μ has the property that for any measure σ whose Fourier-Stieltjes transformσ vanishes at ∞, the measure μ * σ has Radon-Nikodým derivative with respect to λ, the Haar measure on G. Then L. Pigno and S. Saeki showed that μ itself has Radon-Nikodým derivative. Such property is not shared by vector measures in general. We say that a Banach space X has the near differentiability property if every X-valued measure of bounded variation shares the above property. We prove that Banach spaces with the Radon-Nikodým property have the near differentiability property, while Banach spaces with the near differentiability property enjoy the near Radon-Nikodým property. We also show that the Banach spaces L 1 [0, 1] and L 1 /H 1 0 have the near differentiability property. Lastly, we show that Banach spaces with the near differentiability property have type II-Λ-Radon-Nikodým property, whenever Λ is a Riesz subset of type 0 of G.