Math Help - das

1. Finite measure property ?

Hi!
Can any one make this a little bit more clear please?

Let $( E, p )$ be a metric space, and $P:\mathcal B(E)\longrightarrow \mathbb{R}_+$ a probability measure on borel sets of E.

I do not understand the next implication: $x \in E, \epsilon >0$

Since $\delta (B_\epsilon (x)) \subset \{y \in E : p(x,y)=\epsilon \}$
then

$P( B_\epsilon (x))= 0$ for (at most) a numerable number of $\epsilon$

$\delta (B_\epsilon (x))$ is the topological boundary

I think i may be missing a "core" property of finite measures but i dont remember!

$P(B_\epsilon (x))= 0$ EXCEPT for at most a numerable number of $\epsilon$ .