Typical behaviour of integrable functions at infinity

Abstract We obtain the following extension of a theorem due to Lesigne. Let L 1 : = L 1 ( [ 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 c 0 + : = { ( b n ) ∈ c 0 : b n > 0  for all  n ∈ N } . Then, for Y : = L 1 , the sets { ( b , f ) ∈ c 0 + × Y : lim sup n → ∞ f ( n x ) b n = ∞  for almost all  x ≥ 0 } , { f ∈ Y : lim sup n → ∞ f ( n x ) b n = ∞  for almost all  x ≥ 0 } where b ∈ c 0 + , are comeagre of type G δ . If Y : = C ( 1 ) , the analogous sets, with the phrase “for almost all” replaced by “for all”, are also comeagre of type G δ .