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MEASURE, CATEGORY AND CONVERGENT SERIES

机译：测量，类别和会聚系列

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摘要

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.
机译：首先通过斯坦因斯定理及其“对偶”（皮卡德定理）来证明测度和贝儿类别之间的类比。然后，将这两个定理用于根据其收敛子系列集合的“度量大小”和“类别大小”为序列的无条件收敛提供双重准则。作为进一步的应用，在有关可测量和BP可测量功能的基本可分离性的实质性准备部分之后，建立了Cantor组上有关BP_r可测量和普遍可测量加性图的穷举性的结果。在本文的最后部分，研究了关于可测量的通用和BP_r可测量的加性图的可加性的两个经典定理。不仅通过结果本身，而且通过提供的证明来说明所涉及的类比。

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