Exercise 19, § 1

Let $f$ be a scalarly essentially $\mu$-integrable mapping of $T$ into $F$. For every $z' \in F'$, denote by $u_f(z')$ the class in $L^1(\mu)$ of the function $\langle f, z' \rangle$. We shall say that $f$ is scalarly well integrable if, for every function $g \in L^\infty(\mu)$, $\int gf \, d\mu \in F$.

a) For $f$ to be scalarly well integrable, it is necessary and sufficient that $u_f$ be continuous for the topologies $\sigma(F', F)$ and $\sigma(L^1, L^\infty)$. When this is so, the image under $u_f$ of every equicontinuous subset of $F'$ is a relatively weakly compact subset of $L^1$. In particular: $\alpha)$ for every equicontinuous subset $H'$ of $F'$ and every $\varepsilon > 0$, there exists a compact subset $L$ of $T$ such that $\int_{T-L} |\langle f, z' \rangle| \, d\mu \leq \varepsilon$ for all $z' \in H'$; $\beta)$ for every equicontinuous subset $H'$ of $F'$ and every $\varepsilon > 0$, there exists an $\eta > 0$ such that, for every open set $U \subset T$ of measure $\mu(U) \leq \eta$, one has $\int_U |\langle f, z' \rangle| \, d\mu \leq \varepsilon$ for all $z' \in H'$ (Ch. V, §5, Exer. 15).

b) Assume that the conditions $\alpha)$ and $\beta)$ of a) are verified and, moreover, that there exists a subspace $H \subset L^\infty$, dense for the topology $\sigma(L^\infty, L^1)$, such that for every function $g \in \mathscr{L}^\infty$ whose class is in $H$, one has $\int gf \, d\mu \in F$. Show that under these conditions, if in addition $F$ is assumed to be quasi-complete, then $f$ is scalarly well integrable. (First reduce to the case that $F$ is complete (for its original topology), and observe that $g \mapsto \int gf \, d\mu$ is a continuous linear mapping of $L^\infty$ into ${F'}^*$, when $L^\infty$ is equipped with the topology $\tau(L^\infty, L^1)$ and ${F'}^*$ with the topology of uniform convergence on the equicontinuous subsets of $F'$.)

c) Suppose that $f$ is scalarly well integrable. Show that for every Cauchy sequence $(z'_n)$ for the topology $\sigma(F', F)$, the sequence $(u_f(z'_n))$ is strongly convergent in $L^1$ (use Exer. 25 c) of Ch. IV, §5 and Exer. 16 b) of Ch. V, §5). From this, deduce that if $A$ is the image of the unit ball of $L^\infty$ under the mapping $g \mapsto \int gf \, d\mu$, then every Cauchy sequence $(z'_n)$ for the topology $\sigma(F', F)$ is a Cauchy sequence for the topology of uniform convergence in $A$.

d) Deduce from c) that if $f$ is scalarly well integrable, then the image under $u_f$ of every subset of $F'$ that is equicontinuous and metrizable for $\sigma(F', F)$, is relatively strongly compact in $L^1$. In particular, if $F$ is the dual $G'$ of a Hausdorff locally convex space $G$, if $F$ is equipped with a topology compatible with the duality between $F$ and $G$, and if there exists in $G'$ a countable strongly dense subset, then the image under $u_f$ of every bounded subset of $F' = G$ is relatively strongly compact in $L^1$ (embed $G$ in its bidual).