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The Hausdorff measure of the intersection of sets of positive Lebesgue measure
Author(s) -
Erdös P.,
Taylor S. J.
Publication year - 1963
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300003296
Subject(s) - intersection (aeronautics) , mathematics , measure (data warehouse) , hausdorff measure , hausdorff space , lebesgue measure , discrete mathematics , combinatorics , lebesgue integration , hausdorff dimension , computer science , data mining , engineering , aerospace engineering
Erdos, Kestelman and Rogers [1[ showed that, if A 1 , A 2 ,… is any sequence of Lebesgue measurable subsets of the unit interval [0, 1] each of Lebesgue measure at least η > 0, then there is a subsequence {A ni } ( i = 1, 2,…) such that the intersection contains a perfect subset (and is therefore of power ). They asked for what Hausdorff measure functions φ(t) is it possible to choose the subsequence to make the intersection set ∩A ni of positive φ-measure. In the present note we show that the strongest possible result in this direction is true. This is given by the following theorem.