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On two theorems of Sierpiński. (English) Zbl 1401.28002

Let \((X,\mathcal A,\mathcal I)\) be a triple where \(\mathcal I\) is a \(\sigma\)-ideal of subsets of \(X\) and \(\mathcal A\) is a \(\sigma\)-algebra of subsets of \(X\) such that \(\mathcal I\subseteq\mathcal A\). The elements of \(\mathcal A\) and \(\mathcal I\) are called measurable and negligible sets, respectively. Let \(E\subseteq Q\subseteq X\). \(E\) is a full subset of \(Q\), if \(Q\setminus A\in\mathcal I\) for every set \(A\in\mathcal A\) containing \(E\). \(E\) is completely non-measurable in \(Q\), if \(E\) and \(Q\setminus E\) are full subsets of \(Q\). It is assumed that \(\mathcal A\setminus\mathcal I\) satisfies the countable chain condition and that \(X\) has the non-separation property with respect to \(\mathcal A\), i.e., every set \(A\notin\mathcal I\) has disjoint subsets which cannot be separated by a set from \(\mathcal A\). The main result of the paper states that every infinite set \(E\subseteq X\) decomposes into infinite number of disjoint completely non-measurable subsets of \(E\). This result generalizes two theorems of Sierpiński on the Lebesgue measure and the Baire category. The authors present several corollaries strengthening some classical results and discuss some related bibliographical and historical aspects.

MSC:

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54E52 Baire category, Baire spaces
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References:

[1] E. W. Chittenden, Review of [10], Zentralblatt für Mathematik (zbmath.org: document no. 0009.10402). · JFM 48.1254.02
[2] Cichoń, J; Morayne, M; Rałowski, R; Ryll-Nardzewski, C; Żeberski, S, On nonmeasurable unions, Topology Appl., 154, 884-893, (2007) · Zbl 1109.03049 · doi:10.1016/j.topol.2006.09.013
[3] Grzegorek, E, On a paper by karel prikry concerning ulam’s problem on families of measures, Coll. Math., 52, 197-208, (1979) · Zbl 0429.04004 · doi:10.4064/cm-42-1-197-208
[4] E. Grzegorek and I. Labuda, Partitions into thin sets and forgotten theorems of Kunugi and Lusin-Novikov, to appear in Colloq. Math. · Zbl 1428.54006
[5] A. Kumar, On some problems in set-theoretical analysis, PhD Thesis, University of Wisconsin-Madison, 2014, 35 pp., available at http://www.math.huji.ac.il/ akumar/thesis.pdf · Zbl 1207.03056
[6] A. Kumar, Avoiding rational distances, Real Anal. Exchange 38 (2012/13), 493-498. · Zbl 1315.03106
[7] A. Kumar and S. Shelah, A transversal of full outer measure, available at http://www.math.huji.ac.il/akumar/tfom.pdf. · Zbl 1423.03187
[8] Kuratowski, K, Problèmes, Fund. Math., 4, 368-370, (1923) · doi:10.4064/fm-4-1-368-370
[9] K. Kuratowski, Topology, Vol. 1, Academic Press-PWN, 1966. · Zbl 0158.40901
[10] Lusin, N, Sur la décomposition des ensembles, C. R. Acad. Sci. Paris, 198, 1671-1674, (1934) · JFM 60.0039.01
[11] M. Michalski, Odkopane twierdzenie Łuzina, III Warsztaty z Analizy Rzeczywistej, 20-21 May 2017, Konopnica; http://www.im.p.lodz.pl/semwa/pliki/MMichalski2017.pdf. · Zbl 0429.04004
[12] Rałowski, R; Żeberski, S, Completely measurable families, Central European J. Math., 8, 683-687, (2010) · Zbl 1207.03056
[13] A. Rosenthal, Review of [10], Jahrbuch über die Fortschritte der Mathematik (zbmath.org: document no. 60.0039.01).
[14] W. Sierpiński, Hypothèse du continu, Monografje Matematyczne, Warszawa-Lwów, 1934. · JFM 60.0035.01
[15] Sierpiński, W, Sur une propriété des ensembles linéaires quelconques, Fund. Math., 23, 125-134, (1934) · Zbl 0010.01302 · doi:10.4064/fm-23-1-125-134
[16] Żeberski, S, On completely nonmeasurable unions, Math. Log. Quart., 53, 38-42, (2007) · Zbl 1109.03046 · doi:10.1002/malq.200610024
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