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Thinnable Ideals and Invariance of Cluster Points
release_tvz5xwwlqfd4bewcmuebaojcki
by
Paolo Leonetti
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as a article
.
2017
Abstract
We define a class of so-called thinnable ideals I on the positive
integers which includes several well-known examples, e.g., the collection of
sets with zero asymptotic density, sets with zero logarithmic density, and
several summable ideals. Given a sequence (x_n) taking values in a separable
metric space and a thinnable ideal I, it is shown that the set of
I-cluster points of (x_n) is equal to the set of
I-cluster points of almost all its subsequences, in the sense of
Lebesgue measure. Lastly, we obtain a characterization of ideal convergence,
which improves the main result in [Trans. Amer. Math. Soc. 347 (1995),
1811--1819].
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