Lacunary convergence of series in \(L_{0}\). (English) Zbl 0894.46020

Summary: 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.


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
40A30 Convergence and divergence of series and sequences of functions
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
Full Text: DOI


[1] R. P. Agnew, Subseries of series which are not absolutely convergent, Bull. Amer. Math. Soc. 53 (1947), 118-120. · Zbl 0037.04704
[2] H. Auerbach, Über die Vorzeichenverteilung in unendlichen Reihen, Studia Math. 2 (1930), 228-230. · JFM 56.0200.02
[3] L. Drewnowski and I. Labuda, Vector series whose lacunary subseries converge, (1995) (preprint). · Zbl 0949.40001
[4] R. Estrada and R. P. Kanwal, Series that converge on sets of null density, Proc. Amer. Math. Soc. 97 (1986), no. 4, 682 – 686. · Zbl 0592.40001
[5] 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 Loose Russian summary). · Zbl 0164.43002
[6] Dominikus Noll and Wolfgang Stadler, Abstract sliding hump technique and characterization of barrelled spaces, Studia Math. 94 (1989), no. 2, 103 – 120. · Zbl 0711.46004
[7] W. Orlicz, Über die Divergenz von allgemeinen Orthogonalreihen, Studia Math. 4 (1933), 27-32. · JFM 59.1012.01
[8] -, On a class of asymptotically divergent sequences of functions, Studia Math. 12 (1951), 286-307. · Zbl 0044.05701
[9] -, 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.
[10] 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. · Zbl 0496.40008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.