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On the scooping property of measures by means of disjoint balls. (English) Zbl 0892.46052

Let \(X\) be a Banach space. The measures in \(X\) considered below are assumed to be Borel finite measures, and all balls are assumed to be open. We say that a collection \(G\) of subsets of \(X\) has the scooping property \((G\in(\text{s.p.}))\) if for every measure \(\mu\) on \(X\) and for every \(\varepsilon> 0\) there exists a finite family \(\{F_n\}^N_1\subset G\) of pairwise disjoint sets such that \(\mu\left(X\backslash\bigcup^N_{i= 1}F_n\right)< \varepsilon\). This property is related to what is known as the positivity principle (p.p.) introduced by J. P. R. Christensen in “A survey of small ball theorems and problems” [Lect. Notes Math. 794, 24-30 (1980; Zbl 0448.28012)]; the p.p. holds for a family \(G\) of subsets of a Banach space \(X\) if for arbitrary measures \(\mu\) and \(\nu\) on \(X\), the relation \(\mu(B)\geq \nu(B)\) for all \(B\in G\) implies \(\mu\geq \nu\).
The author of this paper mentioned that the family of small balls in the Hilbert space does not possess the scooping property, while D. Preiss [Topology and Measure. III, part 2, 201-207 (1982; Zbl 0515.28009)] has shown before that the same family does not possess the positivity principle. He investigated several properties of the space \(X\) which has not the scooping property. Also, he showed that in the real space \(C[0,1]\) the scooping property holds for the family \(I_{r=1}\) of balls of radius 1. On the contrary, D. Preiss and J. Tiser [Lect. Notes Math. 945, 194-207 (1982; Zbl 0495.28010)] have shown that \(C[0,1]\) has the positivity principle for the same family.
Reviewer: S.Koshi (Sapporo)

MSC:

46G12 Measures and integration on abstract linear spaces
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