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If $mu_1,mu_2,dots$ are positive measures on a measurable space $(X,Sigma)$ and $v_1,v_2, dots$ are elements of a Banach space ${mathbb E}$ such that $sum_{n=1}^infty |v_n| mu_n(X) < infty$, then $omega (S)= sum_{n=1}^infty v_n mu_n(S)$ defines a vector measure of bounded variation on $(X,Sigma)$. We show ${mathbb E}$ has the Radon-Nikodym property if and only if every ${mathbb E}$-valued measure of bounded variation on $(X,Sigma)$ is of this form. As an application of this result we show that under natural conditions an operator defined on positive measures, has a unique extension to an operator defined on ${mathbb E}$-valued measures for any Banach space ${mathbb E}$ that has the Radon-Nikodym property.
We clarify the relation between inverse systems, the Radon-Nikodym property, the Asymptotic Norming Property of James-Ho, and the GFDA spaces introduced in our earlier paper on differentiability of Lipschitz maps into Banach spaces.
This paper deals with a property which is equivalent to generalised-lushness for separable spaces. It thus may be seemed as a geometrical property of a Banach space which ensures the space to have the Mazur-Ulam property. We prove that if a Banach sp
Admissible vectors lead to frames or coherent states under the action of a group by means of square integrable representations. This work shows that admissible vectors can be seen as weights with central support on the (left) group von Neumann algebr
We study the computational content of the Radon-Nokodym theorem from measure theory in the framework of the representation approach to computable analysis. We define computable measurable spaces and canonical representations of the measures and the i
Let $G$ be a topological Abelian semigroup with unit, let $E$ be a Banach space, and let $C(G,E)$ denote the set of continuous functions $fcolon Gto E$. A function $fin C(G,E)$ is a generalized polynomial, if there is an $nge 0$ such that $Delta_{h_1