Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4672728 | Indagationes Mathematicae | 2016 | 9 Pages |
Abstract
We obtain the following extension of a theorem due to Lesigne. Let L1:=L1([0,∞))L1:=L1([0,∞)) and let C(1)C(1) be the (Polish) space of nonnegative continuous functions ff on [0,∞)[0,∞) such that ∫[0,∞)f≤1∫[0,∞)f≤1, with the metric of uniform convergence on every compact subset of [0,∞)[0,∞). Denote c0+:={(bn)∈c0:bn>0 for all n∈N}. Then, for Y:=L1Y:=L1, the sets{(b,f)∈c0+×Y:lim supn→∞f(nx)bn=∞ for almost all x≥0},{f∈Y:lim supn→∞f(nx)bn=∞ for almost all x≥0}whereb∈c0+, are comeagre of type GδGδ. If Y:=C(1)Y:=C(1), the analogous sets, with the phrase “for almost all” replaced by “for all”, are also comeagre of type GδGδ.
Keywords
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Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Marek Balcerzak, Adam Majchrzycki, Filip Strobin,