Real Analysis Exchange

Measure, Category and Convergent Series

Iwo Labuda

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

Subjects
Primary: 40A05: Convergence and divergence of series and sequences 46B15: Summability and bases [See also 46A35]
Secondary: 54C10: Special maps on topological spaces (open, closed, perfect, etc.)

Keywords
Topological group unconditional convergence Baire category Baire property additive map on a Cantor group Haar measure universal measurability.

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


Export citation