Proposition 14

Let $X$ be a locally compact space, $\mu$ a measure on $X$, $A$ a $\mu$-measurable subset of $X$, and $\mathscr{K}$ a set of compact subsets of $A$ that is $\mu$-dense in $A$. There exists a locally countable set $\mathfrak{H} \subset \mathscr{K}$, formed of pairwise disjoint sets, such that $A - \bigcup_{K \in \mathfrak{H}} K$ is locally $\mu$-negligible and such that, for every $K \in \mathfrak{H}$, the support of $\mu_K$ is all of $K$.

Consider sets $\mathscr{L} \subset \mathfrak{K}$ formed of pairwise disjoint sets such that, for every $L \in \mathscr{L}$, $\operatorname{Supp}(\mu_L) = L$. The sets $\mathscr{L}$ form a subset $\mathscr{H}$ of $\mathscr{P}(\mathfrak{K})$ that is nonempty (because it contains the element $\varnothing$) and which we shall order by the relation of inclusion in $\mathscr{P}(\mathfrak{K})$. It is immediate that $\mathscr{H}$ is inductive; let $\mathfrak{H}$ be a maximal element of $\mathscr{H}$ (S, R, §6, No. 10). Let us first show that $\mathfrak{H}$ is locally countable. Indeed, for every $x \in X$, let $V$ be a relatively compact open neighborhood of $x$; if $(K_i)_{1 \leq i \leq n}$ is a finite family of distinct sets of $\mathfrak{H}$ that intersect $V$, then

because the $K_i$ are pairwise disjoint, whence $\sum_{i=1}^n |\mu|(K_i \cap V) \leq |\mu|(V)$. It follows that if $\mathfrak{H}_V$ is the set of $K \in \mathfrak{H}$ that intersect $V$, then

and since $|\mu|(K \cap V) > 0$ for every $K \in \mathfrak{H}_V$, $\mathfrak{H}_V$ is necessarily countable. Next, let us prove that $N = A - \bigcup_{K \in \mathfrak{H}} K$ is locally $\mu$-negligible. We saw above that $N$ is $\mu$-measurable. If $N$ were not locally negligible, it would contain a non-negligible compact set $L_0$, hence (No. 8, Prop. 12) a non-negligible compact set $L \subset L_0$ belonging to $\mathfrak{K}$. Since $|\mu_L|(L) = |\mu|(L) > 0$ (No. 7, Lemmas 2 and 3), the measure $\mu_L$ induced on $L$ by $\mu$ is nonzero; its support $S$ is therefore a nonempty compact set belonging to $\mathfrak{K}$ by (PL₁), and $\operatorname{Supp}(\mu_S) = S$ (No. 7, Lemma 2, (iii)). It follows that the set $\mathfrak{H} \cup \{S\}$ belongs to $\mathscr{H}$, which contradicts the definition of $\mathfrak{H}$; the set $N$ is therefore locally negligible, which completes the proof.