Exercise 9, § 4

a) Let $\mu$ be a positive measure on a locally compact space $X$. In the Banach space $\widetilde{\mathscr{F}}^1$ of equivalence classes of functions in $\mathscr{F}^1$, let $G$ be a closed linear subspace containing the subspace $L^1$; let $\mathscr{G}$ be the closed linear subspace of $\mathscr{F}^1$ formed by the functions whose class belongs to $G$. Show that one can extend to $\mathscr{G}$ the linear form $\mu(f)$ in such a way that the inequality $|\mu(f)| \leq N_1(f)$ is again verified (make use of the Hahn–Banach theorem). Show that for the integral so extended, Th. 3 of §3 is again valid.

b) Assume that $G$ is the direct sum of $L^1$ and a subspace $H$ of finite dimension. Show that, in $\mathscr{G}$, Lebesgue’s theorem (Th. 2) is again valid. (Let $(f_n)$ be a sequence of functions in $\mathscr{G}$, tending almost everywhere to $f$ and such that $|f_n| \leq g$, where $g \geq 0$ and $N_1(g) < +\infty$. Let $(\tilde{u}_k)_{1 \leq k \leq m}$ be a basis of $H$, and let $f_n = \sum_{k=1}^m \alpha_{nk} u_k + h_n$, where $h_n \in \mathscr{L}^1$. Using the fact that $G$ is the topological direct sum of $L^1$ and $H$, show that the $\alpha_{nk}$ are uniformly bounded; on passing, if necessary, to a subsequence of $(f_n)$, reduce to the case that each of the sequences $(\alpha_{nk})_{n \geq 1}$ has a limit; from this, deduce that $h_n(x)$ then tends to a limit almost everywhere, and apply Lebesgue’s theorem to the sequence $(h_n)$; note finally that two subsequences of $(\tilde{f}_n)$ cannot tend to different limits in $G$.)