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Small systems convergence and metrizability. (English) Zbl 0651.28004
A strong small system is a decreasing sequence \((N_ n)_ n\) of nonempty families of subsets of X such that for each integer r \((i)\quad E_ i\in N_ i\quad (i=r+1,r+2,...)\) implies \(\cup^{\infty}_{i=r+1}E_ i\in N_ r\) and \((ii)\quad E,F\in N_{r+1}\) implies \(E\cup F\in N_ r.\)
The authors define the convergence of a sequence of real functions defined on X, what is a generalization of the convergence in measure. Then they show that this convergence is equivalent with the convergence with respect to an appropriate metric. Similar problems have been studied also by J. Komorník [Mat. Čas., Slovensk. Akad. Vied. 25, 59-62 (1975; Zbl 0295.28001)] and O. Kulcsárová and the reviewer [Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14, 385-389 (1987)].
Reviewer: B.Riečan

MSC:
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
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[1] Peter Capek, On small systems, Acta Fac. Rerum Natur. Univ. Comenian. Math. 34 (1979), 93 – 101 (English, with Russian and Slovak summaries). · Zbl 0436.28003
[2] Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna. Tom 47. [Mathematics Library. Vol. 47]. Ryszard Engelking, General topology, PWN — Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60].
[3] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. · Zbl 0040.16802
[4] Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. · Zbl 0060.39808
[5] John L. Kelley, General topology, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. · Zbl 0066.16604
[6] M. E. Munroe, Introduction to measure and integration, Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1953. · Zbl 0050.05603
[7] Tibor Neubrunn and Beloslav Riečan, Miera a integrál, VEDA, Vydavatel’stvo Slovenskej Akadémie Vied, Bratislava, 1981 (Slovak). With Russian and English summaries. · Zbl 0485.28001
[8] Jerzy Niewiarowski, Convergence of sequences of real functions with respect to small systems, Math. Slovaca 38 (1988), no. 4, 333 – 340 (English, with Russian summary). · Zbl 0659.28004
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