Real Analysis Exchange

Measure, Category and Convergent Series

Iwo Labuda

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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

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

First available in Project Euclid: 10 May 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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.)

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


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

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