Proposition 4

— In order that a bounded complex measure $\theta$ on a completely regular space $T$ be positive, it is necessary and sufficient that $\theta(f) \geq 0$ for every function $f \in \mathscr{C}_+^b(T)$.

Necessity is obvious. To establish sufficiency, let us take up again the proof of the preceding proposition, on taking $p = 1$ and $\mu = |\theta|$; the notations being the same, the relation $\mu^*(|f - g|) \leq \varepsilon$ and the inequality $\theta(f) \geq 0$ imply $\theta_K(g_K) = \theta(g) \geq -\varepsilon$; since $g_K$ is an arbitrary element of $\mathscr{C}(K)$ between 0 and 1, the measure $\theta_K$ is positive; the compact set K being arbitrary, this means that $\theta$ is positive.