Proposition 13

Let $A$ be a $\mu$-measurable subset of $X$, $\mathscr{K}$ a set of compact subsets that is $\mu$-dense in $A$. Let $\mathfrak{H}$ be a set of compact subsets of $A$ satisfying (PL$_I$) and (PL$_{II}$) and such that, for every $K \in \mathscr{K}$, the set of $H \in \mathfrak{H}$ such that $H \subset K$ is $\mu_K$-dense* (or, what amounts to the same, $\mu$-*dense*) in $K$. Then $\mathfrak{H}$ is $\mu$-dense in $A$.*

For, let $L$ be a compact subset of $A$. For every $\varepsilon > 0$ there exists a $K \in \mathscr{K}$ such that $K \subset L$ and $|\mu|(L - K) \leq \varepsilon/2$, and then an $H \in \mathfrak{H}$ such that $H \subset K$ and $|\mu|(K - H) \leq \varepsilon/2$; it follows that $|\mu|(L - H) \leq \varepsilon$, whence the proposition.