Lacunary Convergence of Series in L0 Lech Drewnowski and Iwo Labuda

For a finite measure λ, let L0(λ) denote the space of λ-measurable functions equipped with the topology of convergence in measure. We prove that a series in L0(λ) is subseries (or unconditionally) convergent provided each of its lacunary subseries converges. A series in a topological vector space is said to be subseries convergent if each of its subseries converges. As is well known, a subseries convergent series is unconditionally convergent, and the converse holds in sequentially complete spaces. A strictly increasing sequence (nk) in N = {1, 2, . . .}, or the set {n1, n2, . . . }, is called lacunary if limk(nk+1−nk) = ∞, and of density zero if limk(k/nk) = 0. The lacunary subseries (resp. zero-density subseries) of a given series are those corresponding to lacunary sequences of indices (resp. sequences of indices of density zero). In our use of the term ‘lacunary’, we follow [SF]; its Hadamard meaning, e.g. in the theory of trigonometric series, is more restrictive. In 1930, Auerbach [A, Hilfsatz], published the following result which, according to Footnote 1 in his paper, he had already obtained in 1923: (A) A scalar series is subseries (or unconditionally, or absolutely) convergent provided each of its zero-density subseries converges. Without mentioning Auerbach, this result reappeared in a 1986 paper by Estrada and Kanwal [EK, Thm. 1], a 1989 paper by Noll and Stadler [NS, Lemma on p. 116] and, in a stronger form, in a 1980 paper by Sember and Freedman [SF, Prop. 2]. The latter reads as follows: (B) A scalar series is subseries convergent provided each of its lacunary subseries converges. We note that the proofs given by the authors mentioned above are essentially variations of Auerbach’s; indeed, Auerbach’s original proof is also a proof of (B). A natural question arises as to what extent results of types (A) or (B) are valid for series in more general spaces. Received by the editors February 18, 1996. 1991 Mathematics Subject Classification. Primary 46E30, 40A30, 28A20.