Exercise 25, § 5

a) For a subset H of $\mathscr{S}_F$ to be precompact, it is necessary and sufficient that, for every ε > 0 and every integrable set A ⊂ X, there exist a compact set M ⊂ F and a partition of A into a finite number of integrable sets A_i such that, for every f ∈ H, there exists an integrable set B ⊂ A of measure μ(B) ≤ ε and having the following properties: 1° every point of f(A − B) has distance ≤ ε from M; 2° in each of the sets A_i ∩ C B, the oscillation of f is ≤ ε. b) Show that for a subset H of $\mathscr{L}_F^1$ to be relatively quasi-compact, it is necessary and sufficient that it be precompact in $\mathscr{S}_F$ and equi-integrable. c) For a sequence (f_n) of functions in $\mathscr{L}_F^1$ to be a Cauchy sequence for the topology of convergence in mean, show that it is necessary and sufficient that (f_n) be a Cauchy sequence for the topology of convergence in measure and that the set of the f_n be equi-integrable. d) Extend these properties to the spaces $\mathscr{L}_F^p$ for p > 1.