On exposed points of the range of a vector measure. II

Abstract:

If a weakly compact convex set $K$ in a real Banach space $X$ is strongly exposed by a dense set of functionals in $X’$, it is proved that the functionals which expose $K$ form a residual set in $X’$. If $v:\mathcal {A} \to X$ is a measure, it follows that the set of exposing functionals of its range is a residual ${G_\delta }$ in $X’$. This, in turn, is found to be equivalent to a theorem of B. Walsh on the residuality of functionals $x’ \in X’$ for which $x’ \circ v \equiv v$. If the set of exposed points of $v(\mathcal {A})$ is weakly closed and ${v_A}$ is the restriction of $v$ to any set $A \in \mathcal {A}$, it is further proved that every exposed point of the range of ${v_A}$ is of the form $v(A \cap E)$, where $E \in \mathcal {A}$ and $v(E)$ is an exposed point of $v(\mathcal {A})$.