Lemma 1

— Let $K$ be a compact subset of $X$, u a function of $\mathscr{K}_{+}(X)$ with $u(x) > 0$ for $x \in K$. Let $g \in \mathscr{K}^{\chi}(X)$ be such that $\operatorname{Supp} g \subset KH$. a) One has $\inf_{x \in KH} u^{1}(x) > 0$. b) The function $h$ equal to $g/u^{1}$ on $KH$, and to $0$ on $X - KH$, belongs to $\mathscr{K}^{\chi}(X)$. c) $g = (uh)^{\chi}$. One has $u^{1}(x) > 0$ for $x \in K$, therefore $\inf_{x \in KH} u^{1}(x) = \inf_{x \in K} u^{1}(x) > 0$. Assertion b) follows from this at once. Finally, $(uh)^{\chi} = u^{1}h$ by Prop. 1 c), and it is clear that $u^{1}h = g$.

Let I be a relatively bounded linear form (Ch. II, §2, No. 2) on $\mathscr{K}^{\chi}(X)$. Then $f \mapsto I(f^{\chi})$ is a relatively bounded linear form on $\mathscr{K}(X)$, that is, a measure $\mu_{I}$ on $X$. The mapping $I \mapsto \mu_{I}$ is injective by Prop. 2. The measures $\mu_{I}$ on $X$ so obtained may be characterized as follows: