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A-Systems, Independent Functions, and Sets Bounded in Spaces of Measurable Functions
S. Ya. Novikov Samara State University
Abstract:
Let U⊂L∘([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
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
Linking options:
https://www.mathnet.ru/eng/mzm12https://doi.org/10.4213/mzm12 https://www.mathnet.ru/eng/mzm/v75/i1/p115
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Abstract page: | 433 | Full-text PDF : | 170 | References: | 96 | First page: | 1 |
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