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On the convergence of ∑ n = 1 ∞ f ( n x ) for measurable functions
Author(s) -
Buczolich Zoltán,
Mauldin R. Daniel
Publication year - 1999
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/s0025579300007804
Subject(s) - mathematics , function (biology) , measurable function , convergence (economics) , set (abstract data type) , pure mathematics , combinatorics , mathematical analysis , computer science , economics , economic growth , evolutionary biology , bounded function , biology , programming language
Questions of Haight and of Weizsäcker are answered in the following result. There exists a measurable function f: (0, + ∞) → {0,1} and two non‐empty intervals I F I ∞ ⊂[½,1) such that∑ n = 1 ∞ f ( n x )= + ∞for every x εI ∞ , and∑ n = 1 ∞ f ( n x )< + ∞ for almost every x εI f . The function f may be taken to be the characteristic function of an open set E .