Exercise 31, § 5

Let $\theta$ be a complex measure on T, and F a Banach space.

a) Show that the space $L^1_{loc}(T, \theta; F)$ is complete. (Write $|\theta|$ in the form $\sum_\alpha \mu_\alpha$, where $(\mu_\alpha)$ is a summable family of measures $\geqslant 0$ on T whose supports form a locally countable family of pairwise disjoint compact subsets. This defines a continuous mapping of $L^1_{loc}(T, \theta; F)$ into $\prod_\alpha L^1_F(T, \mu_\alpha)$. Let $\mathfrak{F}$ be a Cauchy filter on $L^1_{loc}(T, \theta; F)$. Its image in $\prod_\alpha L^1_F(T, \mu_\alpha)$ converges to an element $(f_\alpha)$. Show that the $f_\alpha$ define a $\theta$-measurable function f on T, then that f is locally $\theta$-integrable, and that $\dot{f}$ is the limit of $\mathfrak{F}$ in $L^1_{loc}(T, \theta; F)$.)

b) Let p be a real number $> 1$. Denote by $\mathscr{L}^p_{loc}(T, \theta; F)$ the vector space of $\theta$-measurable functions f on T with values in F, such that $|f|^p$ is locally $\theta$-integrable. Equip it with the semi-norms $f \mapsto (\int |f \varphi_K|^p d|\theta|)^{1/p}$, where K runs over the set of compact subsets of T. Let $L^p_{loc}(T, \theta; F)$ be the associated Hausdorff space. Show that this space is compete. (Same method as in a).)