Exercise 30, § 5

a) Let M be a bounded subset of $\mathbf{C}$. Show that the closed convex envelope of M is the intersection of the closed disks containing M.

b) Let X be a locally compact space, $\mu$ a complex measure on X such that $\| \mu \| = \mu(1) = 1$, f a bounded, continuous complex function on X. Show that $\mu(f)$ belongs to the closed convex envelope of $f(X)$ in $\mathbf{C}$. (Use a).) Deduce from this a new proof of Prop. 9.