The aim of this paper is to extend the Radon-Nikodym Theorem of Musial (1993) to the case of finitely additive measure’s. The Radon-Nikodym Theorem here proven makes use of the Moedomo-Uhl kind of assumption. The first complication arises from the fact that the finite additivity, due to the lack of the Radon-Nikodym Theorem even in the scalar setting, does not guarantee under the simple assumption of the absolute continuity the existence of the scalar density dx^*nu/ dx^*μ. Hence one has to assume such existence or some suitable conditions ensuring it Moreover, since the proof in the countably additive case is based upon the existence of a lifting, it cannot be mimicked in the present setting; its role in the proof is somehow replaced by assuming that μ admits a Rybakov control. This condition, which is not necessarily satisfied when X is a locally convex topological vector space even for s-bounded μ (as it is when X is a Banach space), is shortly discussed at the end of the paper.
A Radon-Nikodym theorem for the Bartle-Dunford-Schwartz Integral with respect to Finitely Additive Measures
SAMBUCINI, Anna Rita;MARTELLOTTI, Anna;
1994
Abstract
The aim of this paper is to extend the Radon-Nikodym Theorem of Musial (1993) to the case of finitely additive measure’s. The Radon-Nikodym Theorem here proven makes use of the Moedomo-Uhl kind of assumption. The first complication arises from the fact that the finite additivity, due to the lack of the Radon-Nikodym Theorem even in the scalar setting, does not guarantee under the simple assumption of the absolute continuity the existence of the scalar density dx^*nu/ dx^*μ. Hence one has to assume such existence or some suitable conditions ensuring it Moreover, since the proof in the countably additive case is based upon the existence of a lifting, it cannot be mimicked in the present setting; its role in the proof is somehow replaced by assuming that μ admits a Rybakov control. This condition, which is not necessarily satisfied when X is a locally convex topological vector space even for s-bounded μ (as it is when X is a Banach space), is shortly discussed at the end of the paper.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
