Proposition 2

— Let $\eta$ and $\eta'$ be two complex measures on a completely regular space $T$, such that $\eta(f) = \eta'(f)$ for every function $f \in \mathscr{C}^b(T)$ that is integrable for $|\eta|$ and $|\eta'|$. Then $\eta = \eta'$.

Let us take up again the proof of the second part of Proposition 1, on setting $\theta = \eta - \eta'$. We can require the open set U to be integrable for $|\eta|$ and $|\eta'|$. The function $g$ is then integrable for these two measures, and the relation $\theta(g) = 0$ implies $a < 0$; therefore $|\theta|^*(f) = 0$ for every positive lower semi-continuous function $f$, whence finally $|\theta| = 0$, on taking $f = +\infty$.