Lacunary convergence of series in \(L_0\) revisited (Q5926245)

A series in a topological vector space is said to be subseries convergent if each of its subseries converges. A subseries \(\sum_k f_{n_k}\) of a given series \(\sum f_n\) is said to be lacunary if \(n_{k+1}- n_k\to \infty\). For a locally finite measure \(\lambda\), let \(L_0(\lambda)\) denote the topological vector space of all scalar measurable functions equipped with the topology of convergence in measure \(\lambda\) on sets of finite measure. The author proves that a series in \(L_0(\lambda)\) is subseries convergent if each of its lacunay subseries converges. If \(\lambda\) is a finite measure such theorem can be found in joint paper of the author and \textit{I. Labuda} [Proc. Am. Math. Soc. 126, No. 6, 1655-1659 (1998; Zbl 0894.46020)]. In this paper the proof is simpler.