×

zbMATH — the first resource for mathematics

An abstract and generalized approach to the Vitali theorem on nonmeasurable sets. (English) Zbl 07217180
Summary: Here we present abstract formulations of two theorems of S. Solecki [Proc. Am. Math. Soc. 119, No. 1, 115–124 (1993; Zbl 0784.28006); ibid. 897–902 (1993; Zbl 0795.28010)] which deal with some generalizations of the classical Vitali theorem on nonmeasurable sets in spaces with transformation groups.
MSC:
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28D05 Measure-preserving transformations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Erdős, P.; Mauldin, R. D., The nonexistence of certain invariant measures, Proc. Am. Math. Soc. 59 (1976), 321-322
[2] Johnson, R. A.; Niewiarowski, J.; Świątkowski, T., Small systems convergence and metrizability, Proc. Am. Math. Soc. 103 (1988), 105-112
[3] Kharazishvili, A. B., On some types of invariant measures, Sov. Math., Dokl. 16 (1975), 681-684 English. Russian original Translated from Dokl. Akad. Nauk SSSR 222 1975 538-540
[4] Kharazishvili, A. B., Transformations Groups and Invariant Measures, Set-Theoretic Aspects. World Scientific, Singapore (1998)
[5] Niewiarowski, J., Convergence of sequences of real functions with respect to small systems, Math. Slovaca 38 (1988), 333-340
[6] Pelc, A., Invariant measures and ideals on discrete groups, Diss. Math. 255 (1986), 47 pages
[7] Riečan, B., Abstract formulation of some theorems of measure theory, Mat.-Fyz. Čas., Slovensk. Akad. Vied 16 (1966), 268-273
[8] Riečan, B., Abstract formulation of some theorems of measure theory. II, Mat. Čas., Slovensk. Akad. Vied 19 (1969), 138-144
[9] Riečan, B., A note on measurable sets, Mat. Čas., Slovensk. Akad. Vied 21 (1971), 264-268
[10] Riečan, B.; Neubrunn, T., Integral, Measure and Ordering, Mathematics and Its Applications 411. Kluwer Academic Publisher, Dordrecht (1997)
[11] Riečanová, Z., On an abstract formulation of regularity, Mat. Čas., Slovensk. Akad. Vied 21 (1971), 117-123
[12] Solecki, S., Measurability properties of sets of Vitali’s type, Proc. Am. Math. Soc. 119 (1993), 897-902
[13] Solecki, S., On sets nonmeasurable with respect to invariant measures, Proc. Am. Math. Soc. 119 (1993), 115-124
[14] Vitali, G., Sul problema della misura dei gruppi di punti di una retta. Nota, Gamberini e Parmeggiani, Bologna (1905), Italian \99999JFM99999 36.0586.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.