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Matematicheskie Zametki, 2004, Volume 75, Issue 1, Pages 115–134
DOI: https://doi.org/10.4213/mzm12
(Mi mzm12)
 

A-Systems, Independent Functions, and Sets Bounded in Spaces of Measurable Functions

S. Ya. Novikov

Samara State University
References:
Abstract: Let UL([0,1],M,m) be a set of Lebesgue measurable functions. Suppose also that two seminormed spaces of real number sequences are given: A and B. We study (A,B) -sets U defined by the classes A and B as follows:
a=(an)A,(fn(t))uN(or for sequences similar to,(fn(t))E=E(a)[0,1],mE=1such that{anfn(t)1E(t)}B,t[0,1].
We consider three versions of the definition of (A,B) -sets, one of which is based on functions independent in the probability sense. The case B=l is studied in detail. It is shown that (A,l) -independent sets are sets bounded or order bounded in some well-known function spaces (Lp, Lp,q, etc.) constructed with respect to the Lebesgue measure. A characterization of such sets in terms of seminormed spaces of number sequences is given. The (l1,c)- and (A,l1) -sets were studied by E. M. Nikishin.
Received: 01.04.2002
Revised: 28.05.2003
English version:
Mathematical Notes, 2004, Volume 75, Issue 1, Pages 107–123
DOI: https://doi.org/10.1023/B:MATN.0000015026.49971.53
Bibliographic databases:
UDC: 517.5+517.98+519.21
Language: Russian
Citation: S. Ya. Novikov, “A-Systems, Independent Functions, and Sets Bounded in Spaces of Measurable Functions”, Mat. Zametki, 75:1 (2004), 115–134; Math. Notes, 75:1 (2004), 107–123
Citation in format AMSBIB
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\pages 115--134
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\jour Math. Notes
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\pages 107--123
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