Lacunary convergence of series in $L_0$
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- by Lech Drewnowski and Iwo Labuda
- Proc. Amer. Math. Soc. 126 (1998), 1655-1659
- DOI: https://doi.org/10.1090/S0002-9939-98-04189-6
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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
- A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8
- H. Auerbach, Über die Vorzeichenverteilung in unendlichen Reihen, Studia Math. 2 (1930), 228–230.
- L. Drewnowski and I. Labuda, Vector series whose lacunary subseries converge, (1995) (preprint).
- R. Estrada and R. P. Kanwal, Series that converge on sets of null density, Proc. Amer. Math. Soc. 97 (1986), no. 4, 682–686. MR 845987, DOI 10.1090/S0002-9939-1986-0845987-0
- W. Matuszewska and W. Orlicz, A note on modular spaces. IX, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 801–808 (English, with Russian summary). MR 241942
- Dominikus Noll and Wolfgang Stadler, Abstract sliding hump technique and characterization of barrelled spaces, Studia Math. 94 (1989), no. 2, 103–120. MR 1025741, DOI 10.4064/sm-94-2-103-120
- W. Orlicz, Über die Divergenz von allgemeinen Orthogonalreihen, Studia Math. 4 (1933), 27–32.
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- J. J. Sember and A. R. Freedman, On summing sequences of $0$’s and $1$’s, Rocky Mountain J. Math. 11 (1981), no. 3, 419–425. MR 722575, DOI 10.1216/RMJ-1981-11-3-419
Bibliographic Information
- Lech Drewnowski
- Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University, Matejki 48/49, 60–769 Poznań, 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
- 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 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 1655-1659
- MSC (1991): Primary 46E30, 40A30, 28A20
- DOI: https://doi.org/10.1090/S0002-9939-98-04189-6
- MathSciNet review: 1443149