Typical behaviour of integrable functions at infinity

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.