Exercise 14, § 2

a) Let $G$ be a Banach space, $F = G'$ its strong dual, $g$ a mapping of $T$ into $F$ belonging to the space $\Lambda_G^p$ (§1, Exer. 16). Show that if $p > 1$, the measure $g \cdot \mu$ is absolutely continuous with respect to $\mu$ (for the strong topology of $F$).

b) Take for $G$ the Banach space $L^1(\mathbf{N})$, so that $G' = L^\infty(\mathbf{N})$; if $f$ is the function defined in Exer. 7*a*) of §1 (regarded as taking its values in $G'$), then $f$ is scalarly well integrable but the measure $f \cdot \mu$ is not absolutely continuous with respect to $\mu$ for the strong topology of $G'$.