## Real Analysis Exchange

### Measure, Category and Convergent Series

Iwo Labuda

#### 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.

#### Article information

Source
Real Anal. Exchange, Volume 42, Number 2 (2017), 411-428.

Dates
First available in Project Euclid: 10 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.rae/1525917695

Digital Object Identifier
doi:10.14321/realanalexch.42.2.0411

Mathematical Reviews number (MathSciNet)
MR3721809

Zentralblatt MATH identifier
06870337

#### Citation

Labuda, Iwo. Measure, Category and Convergent Series. Real Anal. Exchange 42 (2017), no. 2, 411--428. doi:10.14321/realanalexch.42.2.0411. https://projecteuclid.org/euclid.rae/1525917695