Exercise 18, § 1

Let X be a compact space, $\mu$ a measure on X such that $\mu(1) = \| \mu \|$. Show that $\mu$ is a positive measure. (If $\mu_1 = \mathscr{R}(\mu)$, $\mu_2 = \mathscr{I}(\mu)$, note first that $\mu_1(1) = \| \mu_1 \|$ and show that $\mu_1$ is a positive measure by making use of Ch. II, §2, No. 2, Prop. 4; then prove that $\mu_2 = 0$ by considering $\mu(1 + if)$ for a suitable function $f \in \mathscr{C}(X; \mathbf{R})$.)