Proposition 3

*Let $T$ be a locally compact space, and let $\mu$ be a measure on $T$. Denote by $W(\mu)$ the mapping that associates to each compact subset $K$ of $T$ the induced measure $\mu_K$. Then $W(\mu)$ is a premeasure on $T$, one has $W(|\mu|) = |W(\mu)|$, and the linear mapping $W : \mu \mapsto W(\mu)$ is a bijection of the space $\mathscr{M}(T; \mathbf{C})$ onto the space $\mathscr{P}(T; \mathbf{C})$ of premeasures on $T$. Moreover, if $\mu$ is positive then $\mu^* = (W(\mu))^*$.

It is clear that $W(\mu)$ is a premeasure (Ch. V, §7, No. 2, Prop. 4) and that the mapping $W$ is linear. The relation $W(\mu) = 0$ means that $\mu$ induces the measure 0 on every compact set in $T$; then $\mu(f) = 0$ for $f \in \mathscr{H}(T; \mathbf{C})$, thus $\mu = 0$, which proves that $W$ is injective. It remains to prove that $W$ is surjective. Since every premeasure is a linear combination of positive premeasures, it will suffice to construct, for every positive premeasure $w$, a positive measure $\mu$ such that $w = W(\mu)$. Let $f \in \mathscr{H}(T)$ be a given function, and let $L$ be a compact set containing the support of $f$; the number $w_L(f_L)$ is independent of the choice of $L$, by the definition of induced measures, so that one can set $\mu(f) = w_L(f_L)$; then $\mu$ is a positive linear form on $\mathscr{H}(T)$, that is, a positive measure. Let us verify that $w = W(\mu)$; first, the relation $\mu^\bullet(f) = w_L^\bullet(f_L)$ extends to the case that $f$ is a finite upper semi-continuous function that is positive and is zero outside $L$. For, let $M$ be a compact neighborhood of $L$, and $\mathscr{H}$ the (decreasing directed) set of continuous functions on $T$, with support contained in $M$, that are $\geq f$. Then (Ch. IV, §4, No. 4, Cor. 2 of Prop. 5)

and on the other hand $w_M^\bullet(f_M) = w_L^\bullet(f_L)$ since $f_M$ is zero on $M - L$ (Ch. V, §7, No. 1, Prop. 1). In particular, if $f$ is taken to be the extension by 0 of an element of $\mathscr{H}_+(L)$, this formula shows that $\mu_L = w_L$ by the definition of induced measures, thus indeed $W(\mu) = w$.

If $\mu$ is positive, then

for every $f \in \mathscr{F}_+(T)$ (Ch. V, §1, Def. 1 and §7, Prop. 1). The relation $|W(\mu)| = W(|\mu|)$ is obvious (Ch. IV, §5, No. 7, Lemma 3).

Q.E.D.

When $T$ is locally compact, we shall from now on identify the spaces $\mathscr{M}(T; \mathbf{C})$ and $\mathscr{P}(T; \mathbf{C})$ by means of the bijection $W$.