Exercise 16, § 2

Consider the mappings $u_m$ of $T = [0, 1]$ into a separable Hilbert space, defined in §1, Exer. 1, and set $f_k = u_1 + \cdots + u_k$; show that the vectorial measures $f_k \cdot \mu$ converge uniformly on every bounded subset of $C(T)$ to a vectorial measure $m$ with values in $F$. Moreover, $m$ may be extended by continuity to the space $L^2(\mu)$ and, for every function $f \in L^2$, one has $\int f dm \in F$ and $|\int f dm| \leq N_2(f)$ in $F$; in particular, $m$ is scalarly of base $\mu$ and absolutely continuous with respect to $\mu$ for the strong topology (Exer. 12 c)). However, $m$ is not of base $\nu$ for any positive measure $\nu$ on $T$, and a fortiori (No. 5, Cor. 4 of Th. 1) is not majorizable. (Reduce to the case that $\nu$ has base $\mu$, using Th. 3 of Ch. V, §5, No. 7.)