Measure, Category and Convergent Series

Abstract

The analogy between measure and Baire category is displayed first by a theorem of Steinhaus and its “dual,” a theorem of Piccard. These two theorems are then applied to provide a double criterion for the unconditional convergence of a series in terms of the “measure size” and the “category size“ of the set of its convergent subseries. As a further application, after a substantial preparatory section concerning essential separability of measurable and \(BP\)-measurable functions, the results about exhaustivity of \(BP_r\)-measurable and universally measurable additive maps on the Cantor group are established. In the last sections of the paper, two classical theorems about countable additivity of the universal measurable and \(BP_r\)-measurable additive maps are examined. The analogy in question is illustrated not only by the results themselves, but also by the proofs provided.