Exercise 28, § 5

Let X be a locally compact space, μ a positive measure on X, and P the set of numerical functions (finite or not) f ≥ 0 defined on X and μ-measurable. Let λ be a function defined on P with values ≥ 0 (finite or not) that is positively homogeneous, increasing and convex (cf. Ch. I, No. 1) and, moreover, such that: 1° λ(f) = 0 for every μ-negligible function f ≥ 0; 2° for every increasing sequence (f_n) of functions in P, λ(\sup_n f_n) = \sup_n λ(f_n). For every Banach space F, denote by $\mathscr{L}_F$^λ the set of μ-measurable mappings f of X into F such that λ(|f|) is finite. Show that λ(|f|) is a semi-norm on $\mathscr{L}_F$^λ, and that $\mathscr{L}_F$^λ is complete for the topology defined by this semi-norm.