Erdős, Pál; Kestelman, H.; Rogers, C. A. An intersection property of sets with positive measure. (English) Zbl 0122.29903 Colloq. Math. 11, 75-80 (1963). 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 Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 1 Document MSC: 28A12 Contents, measures, outer measures, capacities Keywords:differentiation and integration, measure theory PDFBibTeX XMLCite \textit{P. Erdős} et al., Colloq. Math. 11, 75--80 (1963; Zbl 0122.29903) Full Text: DOI EuDML Link