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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.

MSC:

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
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