Exercise 20, § 8

a) Let $\nu$ be a positive measure on the unit circle $U : |z| = 1$ in $\mathbf{C}$, and let $\mu$ be the complex measure $j \cdot \nu$, where $j : z \mapsto z$ is the canonical injection of $U$ into $\mathbf{C}$, so that $|\mu| = \nu$. Let $\rho$ be the inverse image of $\nu$ under the canonical local homeomorphism $\varphi : t \mapsto e^{it}$ of $\mathbf{R}$ onto $U$ (§ 6, No. 6). Let $A$ be a $\nu$-measurable subset of $U$; if $\mu(A) = |\mu(A)|e^{i\omega}$, show that $|\mu(A)| \leq \left| \int_{\omega-\pi/2}^{\omega+\pi/2} \Re(e^{i(\theta-\omega)})\ d\rho(\theta) \right|$.

b) Deduce from a) that, as $A$ runs over the set of $\mu$-measurable subsets of $U$, one has

and conclude therefrom that (evaluate the last integral with the help of the Lebesgue–Fubini theorem).

c) Show that the inequality (*) is true for every bounded complex measure on a locally compact space X, where A runs over the set of $\mu$-measurable subsets of X. (Let $\mu = h \cdot |\mu|$, where h takes its values in U. Consider the image measure $\nu = h(|\mu|)$ on U and show that $h(\mu) = j \cdot \nu$; then apply b) to $h(\mu)$.)