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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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##### References:
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