Let S be the set of subsequences (xnk) of a given real sequence (x n ) which preserve the set of statistical cluster points. It has been recently shown that S is a set of full (Lebesgue) measure. Here, on the other hand, we prove that S is meager if and only if there exists an ordinary limit point of (x n ) which is not also a statistical cluster point of (x n ). This provides a non-analogue between measure and category.
Duality between Measure and Category of Almost All Subsequences of a Given Sequence
Leonetti P
;
2019-01-01
Abstract
Let S be the set of subsequences (xnk) of a given real sequence (x n ) which preserve the set of statistical cluster points. It has been recently shown that S is a set of full (Lebesgue) measure. Here, on the other hand, we prove that S is meager if and only if there exists an ordinary limit point of (x n ) which is not also a statistical cluster point of (x n ). This provides a non-analogue between measure and category.
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