Conglomerated filters and statistical measures

Kadets, Vladimir; Seliutin, Dmytro; Tryba, Jacek

doi:10.1016/j.jmaa.2021.125955

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Hence It Belongs To Ao S(i)1

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2023

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“…(Using [24, Proposition 2.1], one can construct an ideal with the Baire property which does not have the hereditary Baire property). On the other hand, there is no use defining the hereditary Borel ideals because it is known that if I is a Borel ideal then I C is a Borel ideal for every C ∈ I + (see for instance the proof of [15,Theorem 3.13]). Consequently, Borel ideals have the hereditary Baire property as well as they are hereditary Lebesgue'a measurable.…”

Section: Hence It Belongs To Ao S(i)mentioning

confidence: 99%

The ideal test for the divergence of a series

Filipów

Kwela

Tryba

2023

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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We generalize the classical Olivier’s theorem which says that for any convergent series $$\sum _n a_n$$ ∑ n a n with positive nonincreasing real terms the sequence $$(n a_n)$$ ( n a n ) tends to zero. Our results encompass many known generalizations of Olivier’s theorem and give some new instances. The generalizations are done in two directions: we either drop the monotonicity assumption completely or we relax it to the monotonicity on a large set of indices. In both cases, the convergence of $$(na_n)$$ ( n a n ) is replaced by ideal convergence. In the second part of the paper, we examine families of sequences for which the assertions of our generalizations of Olivier’s theorem fail. Here, we are interested in finding large linear and algebraic substructures in these families.

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Section: Hence It Belongs To Ao S(i)mentioning

confidence: 99%