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ISSN 1088-6826(e) ISSN 0002-9939(p)

Lacunary convergence of series in $L_{0}$

Author(s): Lech Drewnowski; Iwo Labuda
Journal: Proc. Amer. Math. Soc. 126 (1998), 1655-1659.
MSC (1991): Primary 46E30, 40A30, 28A20
MathSciNet review: 1443149
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Abstract: For a finite measure $\lambda$ , let $L_{0}(\lambda )$ denote the space of $\lambda$ -measurable functions equipped with the topology of convergence in measure. We prove that a series in $L_{0}(\lambda )$ is subseries (or unconditionally) convergent provided each of its lacunary subseries converges.

References:

[Ag]: R. P. Agnew, Subseries of series which are not absolutely convergent, Bull. Amer. Math. Soc. 53 (1947), 118-120. MR 8:456c
[A]: H. Auerbach, Über die Vorzeichenverteilung in unendlichen Reihen, Studia Math. 2 (1930), 228-230.
[DL]: L. Drewnowski and I. Labuda, Vector series whose lacunary subseries converge, (1995) (preprint).
[EK]: R. Estrada and R. P. Kanwal, Series that converge on sets of null density, Proc. Amer. Math. Soc. 97 (1986), 682-686. MR 87i:40001
[MO]: W. Matuszewska and W. Orlicz, A note on modular spaces.IX, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys. 16 (1968), 801-808. MR 39:3278
[NS]: D. Noll and W. Stadler, Abstract sliding hump technique and characterization of barrelled spaces, Studia Math. 94 (1989), 103-120. MR 91a:46006
[O1]: W. Orlicz, Über die Divergenz von allgemeinen Orthogonalreihen, Studia Math. 4 (1933), 27-32.
[O2]: -, On a class of asymptotically divergent sequences of functions, Studia Math. 12 (1951), 286-307. MR 13:936d
[O3]: -, O szeregach doskonale zbie\.{z}nych w pewnych przestrzeniach funkcyjnych, Prace Mat. 1 (1955), 393-414; English transl., On perfectly convergent series in certain function spaces, W. Orlicz, Collected Papers, Part I, Polish Scientific Publishers, Warszawa, 1988, pp. 830-850. MR 17:511c
[SF]: J. J. Sember and A. R. Freedman, On summing sequences of $0$ 's and $1$ 's, Rocky Mountain J. Math. 11 (1981), 419-425. MR 84k:40006

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Additional Information:

Lech Drewnowski
Affiliation: Faculty of Mathematics and Computer Science, A.~ Mickiewicz University, Matejki 48/49, 60--769 Poznan, Poland
Email: drewlech@math.amu.edu.pl

Iwo Labuda
Affiliation: Department of Mathematics, University of Mississippi, University, Mississippi 38677
Email: mmlabuda@vm.cc.olemiss.edu

DOI: 10.1090/S0002-9939-98-04189-6
PII: S 0002-9939(98)04189-6
Keywords: Space of all measurable functions, subseries convergence in measure, lacunary subseries
Received by editor(s): February 18, 1996
Additional Notes: The final version of this paper was written while the first author held a visiting position in the Department of Mathematics, University of Mississippi, in the Spring Semester of 1996. The first author was also partially supported by the State Committee for Scientific Research (Poland), grant no. 2 P301 003 07.
Communicated by: Dale Alspach
Copyright of article: Copyright 1998, American Mathematical Society

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