Radon-Nikodym性质的几个充要条件

Several Necessary and Sufficent Conditions on Radon-Nikodym Property

本文证明了下列结果:Banach空间x具有Radon-Nikodym性质当且仅当对于任何可测空间(Ω,Σ)。对于任何可数可加有界变差的向量测度α:→X,α关于|α|有Radon-Nikodym导数;当且仅当对于任何α-有限的测度空间(Ω,Σ,μ),X关于(Ω,Σ,μ)具有Radon-Nikodym性质;当且仅当y(x)具有Radon-Nikodym性质。

It is proved in this paper that each of the following conditions is necessary and sufficent for a Banach space X to have the Radon-Nikodym property. They are: (i) for any meadusable space (Ω,Σ) and any vector measuse of countable, additive and bounded variation G:Σ→X, G has Radon-Nikodym derivate in respect to |G| on Σ; (ii) for any σ-finite measure space (Ω,Σ,μ), X has the Radon-Nikodym property in respect to (Ω,Σ,μ); (iii) when and only when μ(x) has the Radon-Nikodym property.