property of borel measures I am reading chapter 1.3 in weak convergence and empirical processes of Van der Vaart and Wellner. Let $(\mathbb{D}, d)$ a metric space and let $L$ a Borel probability measure. In the proof of the Portmanteau theorem they use following fact which I don’t understand. Let $F \subset \mathbb{D},$ be closed. Write $F^\epsilon = \\{ x: d(x,F) < \epsilon \\}$. The set $\delta F^\epsilon$ (= the boundary of $F^\epsilon$) is disjoint for different values of $\epsilon > 0$ (This is OK) , so that at most countably many of them can have nonzero $L$-measure . Can anyone explain to me why there exist at most countably many of them which have nonzero $L$-measure?

For $k \in \mathbb{N}$, define $A_k = \\{ \epsilon > 0: L(\delta F^{\epsilon}) > \frac{1}{k} \\}$. Notice that $A:= \bigcup_k A_k = \\{ \epsilon > 0: L(\delta F^{\epsilon}) > 0 \\}$.

Suppose $A$ is uncountable. Then at least one of the $A_k$ must be uncountable and in particular infinite. But if some $A_k$ is infinite, then by disjointness of the $\delta F^{\epsilon}$, $L(\mathbb{D})$ would be infinite, which is a contradiction.