## 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

### Keywords:

scooping property; positivity principle

### Citations:

Zbl 0448.28012; Zbl 0515.28009; Zbl 0495.28010
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