Proposition 9

— Let $\mu$ be a positive measure on $X$; if $f$ is a function in $\mathscr{K}_+(X)$ such that $\mu(f) = 0$, then $f$ is zero on $\mathrm{Supp}(\mu)$.

Let $x$ be a point of $X$ such that $f(x) > 0$; let us show that $x$ does not belong to $\mathrm{Supp}(\mu)$. For, there exist a compact neighborhood $V$ of $x$ and a number $a > 0$ such that $f(y) \geq a$ on $V$. If $g$ is any continuous function $\geq 0$ with support contained in $V$, let us show that $\mu(g) = 0$; indeed, if one sets $b = \|g\|$ then $g \leq bf/a$, whence $\mu(g) \leq b\mu(f)/a = 0$.