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An intersection property of sets with positive measure. (English) Zbl 0122.29903

The main theorem of this paper runs as follows: ”Let \(X\) be a compact set. Suppose the topology in \(X\) has a countable base. Let \(\mu\) be a Carathéodory outer measure on \(X\) with the properties: (a) \(\mu(X) = 1\), (b) \(\mu(\{x\}) = 0\) for each \(x\) in \(X\), (c) Borel sets in \(X\) are \(\mu\)-measurable, (d) if \(E\) is \(\mu\)-measurable and \(\varepsilon > 0\), then there is an open set \(G\) with \(E \subset G\) and \(\mu(G) < \mu(E) + \varepsilon\). Suppose \(\eta > 0\) and \(A_r\), \(r \in N\), are \(\mu\)-measurable subsets of \(X\) with \(\limsup \mu(A_r) \geq \eta\). Then there is a Borel set \(S\) in \(X\) with \(\mu(S) \geq \eta\), and a sequence \(q_1 < q_2 < ...,\) such that every point of \(S\) is a point of condensation of the set \(\cup_{i \geq 1} \cap_{r \geq i} A_{q_r}\), and every open set containing a point of \(S\) also contains a perfect subset of \(\cap_{i=0} A_{q_{j+i}}\) for some \(j\)”.
Reviewer: P.Georgiou

MSC:

28A12 Contents, measures, outer measures, capacities
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