Typical behaviour of integrable functions at infinity

Abstract

We obtain the following extension of a theorem due to Lesigne. Let L1:=L1([0,∞)) and let C(1) be the (Polish) space of nonnegative continuous functions f on [0,∞) such that ∫[0,∞)f≤1, with the metric of uniform convergence on every compact subset of [0,∞). Denote c0+:={(bn)∈c0:bn>0 for all n∈N}. Then, for Y:=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 Gδ. If Y:=C(1), the analogous sets, with the phrase “for almost all” replaced by “for all”, are also comeagre of type Gδ.