Remark

— Suppose that $A$ is the union of a sequence $(L_n)$ of compact sets and a $\mu$-negligible (resp. locally $\mu$-negligible) set, and let $\mathscr{K}$ be a set of compact subsets that is $\mu$-dense in $A$. Applying to each $L_n$ the property c) of the statement of Prop. 12, we see that $A$ is the union a sequence of compact sets belonging to $\mathscr{K}$ and a $\mu$-negligible (resp. locally $\mu$-negligible) set.

If $K$ is a compact subset of $X$, it comes to the same to say that a set of compact subsets of $K$ is $\mu$-dense in $K$ or that it is $\mu_K$-*dense* in $K$; this follows from Lemmas 2 and 3 of No. 7 and condition b) of Prop. 12.