Exercise 6, § 1

Let $\mu$ be a measure on a locally compact space X. Show that $|\mu| = \sup \mathscr{R}(\alpha \mu)$, where the scalar $\alpha$ runs over the set of complex numbers of absolute value $\leq 1$. (Observe that for every function $g \in \mathscr{K}(X; \mathbf{C})$ and every $\varepsilon > 0$, there exist a finite number of mappings $h_i$ of X into $[0,1]$, with compact supports, and scalars $\alpha_i$ such that $\|g - \sum_i \alpha_i h_i\| \leq \varepsilon$ and $\| |g| - \sum_i |\alpha_i| h_i \| \leq \varepsilon .$)

Show that also $|\mu| = \sup \mathscr{R}(h \cdot \mu)$, where $h$ runs over the set of functions in $\mathscr{H}(X; \mathbf{C})$ such that $\|h\| \leq 1$.