Exercise 15, § 2

a) Let $f$ be a scalarly locally integrable mapping of $T$ into a quasi-complete Hausdorff locally convex space $F$. Then $\mathbf{m} = f \cdot \mu$ is a vectorial measure with values in ${F'}^*$ (for the topology $\sigma({F'}^*, F')$). Show that if this measure is absolutely continuous (for the topology of uniform convergence on the equicontinuous subsets of $F'$), and if, moreover, $f$ is $\mu$-*measurable* (for the original topology of $F$), then $\mathbf{m}$ is a vectorial measure with values in $F$. (Make use of Prop. 8 of §1, No. 2.) If $F$ is a Banach space and if, for some $p > 1$, $\langle z', f \rangle$ belongs to $\overline{\mathscr{L}}^p(\mu)$ for all $z' \in F'$, then the measure $\mathbf{m}$ is absolutely continuous (cf. Exer. 14 a)).

b) Let $I = [0, 1]$, $f$ the function defined on $I \times I$ (using the continuum hypothesis) such that, for every $t \in I$, $s \mapsto f(s, t)$ is the characteristic function of a countable set, and, for every $s \in I$, $t \mapsto f(s, t)$ is the characteristic function of the complement of a countable set (Ch. V, §8, Exer. 7 c)). Denote by $I_0$ the interval $I$ equipped with the discrete topology, by $F$ the Banach space $\mathscr{B}(I) = L^\infty(I)$ of bounded functions on $I$; one can identify $F$ with the space $\mathscr{C}(\widetilde{I}_0)$ of continuous functions on the Stone–Čech compactification of $I_0$. For every $s \in I$, denote by $g(s)$ the element $t \mapsto f(s, t)$ of $F$; show that $g$ is scalarly integrable for the Lebesgue measure $\mu$ on $I$; but $\int g\, d\mu$ does not belong to $F$. More precisely, $g \cdot \mu$ is a vectorial measure with values in $F''$, equal to $c\mu$, where $c$ is the constant vector identified with the characteristic function of $\widetilde{I}_0 - I_0 = E$ (cf. Exer. 12 a)). (Decompose every positive measure on $\widetilde{I}_0$ as the sum of an atomic measure and a diffuse measure (Ch. V, §5, No. 10, Prop. 15), and observe that the diffuse measure is carried by E.) From this, deduce that there does not exist any $\mu$-measurable function $h$ with values in $F$ such that $h - g$ is scalarly $\mu$-negligible (make use of $a$).