Atomic decomposition of vector measures
Let F be a K-space [cf. \textit{L. V. Kantorovich, B. Z. Vulikh} and \textit{A. G. Pinsker}, Usp. Mat. Nauk 6, Nr. 3(43), 31-98 (1951; Zbl 0043.332); English translation in Amer. Math. Soc., Transl., II. Ser. 27, 51-124 (1963; Zbl 0125.341)]. The paper deals with an atomic structure of the set F-bva(\({\mathcal A},Y)\) of measures (finitely additive) with bounded F- variation from an arbitrary Boolean algebra \({\mathcal A}\) into an F-Banach space Y. For \(\mu\in F-bva({\mathcal A},Y)\) the F-variation \(| \mu |\) of \(\mu\) is defined by \[ | \mu | (a):=\sup \{\sum^{n}_{k=1}| \mu (a_ k)|:\quad a_ k\in {\mathcal A}, \] \(k=1,2,...,n\) mutually disjoint and \(a=a_ 1\vee a_ 2\vee...\vee a_ n\},\) for every \(a\in {\mathcal A}\). Let B be a particular Boolean algebra of projections in F. It is shown that the set F-bva(\({\mathcal A},Y)\) is an F-Banach space under the norm \(\mu\) \(\to | \mu |\). In particular an element \(\mu \in F- bva({\mathcal A},Y)\) is called F-singular if its variation \(| \mu |\) is disjoint with every positive measure \(\nu: {\mathcal A}\to F\). Among several results \(\mu\) is F-singular if and only if \(\mu\) cannot be represented as a sum of two disjoint elements in \(F-bva({\mathcal A},Y)\) such that each is of the form \(a\to h(a)y\), \(a\in {\mathcal A}\), \(0\neq y\in Y\) with \(h: {\mathcal A}\to B\) a Boolean isomorphism between \({\mathcal A}\) and B. Also \(\mu\) is F-singular if and only if for every \(e\in F^+\) and \(b\in B\) with \(be\neq 0\) there exists a nonzero projection \(b_ 0\leq b\) and a disjoint, finite subset \(\{\mu_ 1,\mu_ 2,...,\mu_ n\}\subseteq F- bva({\mathcal A},Y)\) such that \(\mu =\mu_ 1+...\mu_ n\) and \(b_ 0| \mu_ k| (1)\leq e,\) for \(k=1,2,...,n.\) Moreover, given \(\mu \in F- bva({\mathcal A},Y)\) there exist an F-singular measure \(\mu_ 0\in F- bva({\mathcal A},Y),\) a sequence \((\mu_ n)\) from disjoint isomorphisms from \({\mathcal A}\) into B, and a sequence \((y_ n)\) in Y such that \(| y_{n+1}| \leq | y_ n|\) and \(\mu(a)=\mu_ 0(a)+\sum^{\infty}_{n=1}\mu_ n(a)y_ n,\) for every \(a\in {\mathcal A}\) where the series \(\sum^{\infty}_{n=1}| y_ n|\) o-converges.
scientific article; zbMATH DE number 4146319