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Sequences of generous and nonatomic quasi-measures on Boolean algebras. (English) Zbl 0993.28002

Let \(A\) be a Boolean algebra. A function \(\nu : A\to[0,\infty]\) is called a quasi-measure if it is additive and satisfies \(\nu(0)=0\) and \(\nu(1)>0\). A quasi-measure \(\nu\) is generous if \(\nu(1)=\infty\) and if each \(a\in A\) with \(\nu(a)=\infty\) can be written as \(a_1\vee a_2\) with \(a_1 \wedge a_2=0\) and \(\min\{\nu(a_1),\nu(a_2)\}=\infty\); it is nonatomic if each \(a \in A\) with \(\nu(a)>0\) can be written as \(a_1 \vee a_2\) with \(a_1 \wedge a_2=0\) and \(\min\{\nu(a_1),\nu(a_2)\} >0\). The notion of a generous quasi-measure is new. For a sequence \(\{\nu_i\}\) of quasi-measures which are all generous, resp. nonatomic, the author studies the existence of a disjoint sequence \(\{a_j\}\) satisfying \(\nu_i(a_i)=\infty\), resp. \(>0\), for all \(i\) or \(\nu_i(a_j)=\infty\), resp. \(>0\), for all \(i\), \(j\). The results are applied to countably additive vector measures of infinite variation in a normed vector space.

MSC:

28A12 Contents, measures, outer measures, capacities
28A60 Measures on Boolean rings, measure algebras
28B05 Vector-valued set functions, measures and integrals
06E99 Boolean algebras (Boolean rings)
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