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Simon Mukwembi 《Quaestiones Mathematicae》2016,39(5):577-585
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《Quaestiones Mathematicae》2013,36(6):765-779
AbstractAssume that is an ideal on ?, and ∑n xn is a divergent series in a Banach space X. We study the Baire category, and the measure of the set A() := {t ∈ {0, 1}?: ∑n t(n)xn is -convergent}. In the category case, we assume that has the Baire property and ∑n xn is not unconditionally convergent, and we deduce that A() is meager. We also study the smallness of A() in the measure case when the Haar probability measure λ on {0, 1}? is considered. If is analytic or coanalytic, and ∑n xn is -divergent, then λ(A()) = 0 which extends the theorem of Dindo?, ?alát and Toma. Generalizing one of their examples, we show that, for every ideal on ?, with the property of long intervals, there is a divergent series of reals such that λ(A(Fin)) = 0 and λ(A()) = 1. 相似文献
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