On Marczewski-Burstin like characterizations of certain \(\sigma\)-algebras and \(\sigma\)-ideals. (English) Zbl 1009.28002

Summary: Consider a \(\sigma\)-ideal, \(\sigma\)-algebra pair \({\mathcal I}\subseteq{\mathcal A}\) on a Polish space \(X\) which has no isolated points, such that \({\mathcal A}\) contains all the Borel subsets of \(X\) while \({\mathcal I}\) contains all the countable subsets of \(X\), but none of the perfect subsets of \(X\). We show that if \(({\mathcal I},{\mathcal A})\) admits a simultaneous Marczewski-Burstin-like characterization consisting of Borel sets, then \(({\mathcal I},{\mathcal A})\) is \(((s_0),(s))\), the \(\sigma\)-ideal, \(\sigma\)-algebra pair of Marczewski null, Marczewski measurable sets. We deduce some results about uniformly completely Ramsey sets.


28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets