Proposition 5

— *Let T be a completely regular space, and I a continuous complex linear form on the normed space $\mathscr{C}^b(T; \mathbf{C})$. In order that there exist a bounded complex measure $\theta$ on T such that $\theta(f) = I(f)$ for all $f \in \mathscr{C}^b(T; \mathbf{C})$, it is necessary and sufficient that the following condition be satisfied:

(M) *For every number $\varepsilon > 0$, there exists a compact subset K of T such that the relations $g \in \mathscr{C}^b(T; \mathbf{C}), |g| \leq 1, g_K = 0$ imply $|I(g)| \leq \varepsilon$.

The measure $\theta$ is then unique.*

Uniqueness follows from Prop. 2 of No. 1. Let us show that the condition (M) is necessary. Let $\theta$ be a bounded complex measure; let K be a compact set such that $|\theta|^*(T - K) \leq \varepsilon$ (§ 1, No. 2, Remark 3). The hypotheses $|g| \leq 1, g_K = 0$ imply $|g| \leq \varphi_{\mathbf{C}_K}$, therefore $|\theta(g)| \leq |\theta|^*(\varphi_{\mathbf{C}_K}) \leq \varepsilon$.

Let us pass to the proof of sufficiency. Let X be the Stone–Čech compactification of T (GT, IX, § 1, Exer. 7; or TG, IX, § 1, No. 6). For every function $f \in \mathscr{C}(X; \mathbf{C})$, set $\nu(f) = I(f_T)$; we define in this way a continuous linear form on $\mathscr{C}(X; \mathbf{C})$, that is, a complex measure on the compact space X. Let $\varepsilon$ be a number $> 0$, K a compact set satisfying (M); the function $\varphi_{\mathbf{C}_K}$ being lower semi-continuous and positive on X, the formula (2) gives us the following relations, where $\mathscr{G}$ denotes the set of functions $g \in \mathscr{C}(X; \mathbf{C})$ such that $|g| \leq \varphi_{\mathbf{C}_K}$:

Let $(K_n)_{n \geq 1}$ be a sequence of compact subsets of T, such that each $K_n$ satisfies (M) for $\varepsilon = 1/n$, and let $S = \bigcup K_n$; S is a Borel set in X, contained in T, and $|\nu|^*(X - T) \leq |\nu|^*(X - S) \leq |\nu|^*(X - K_n) \leq 1/n$ for all n, so that T is $\nu$-measurable and $\nu$ is concentrated on T. Let $f$ be a bounded continuous function on T; since X is the Stone–Čech compactification of T, $f$ may be extended by continuity to a function $g \in \mathscr{C}(X; \mathbf{C})$. Now let $\mu$ be the measure induced by $\nu$ on $T$; one has $\mu(f) = \nu(f^0)$. (1) Since $\nu$ is concentrated on $T$, the functions $f^0$ and $g$ are equal $\nu$-almost everywhere, therefore $\mu(f) = \nu(g) = I(g_T) = I(f)$, which completes the proof.