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 transform σ ˆ 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 0 1 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 ˆ .