Exercise 22, § 5

Show that if the positive measure μ is such that there exist open sets of arbitrarily small measure > 0 then, for every a > 0, the topology induced on the set of f ∈ $\mathscr{L}_F^p$ such that N_p(f) ≤ a by the topology of convergence in measure is strictly coarser than the topology of convergence in mean of order p.