Does the set of limit points always have measure zero?

Question

Let $P$ be a Borel set, does the following set have measure zero(w.r.t to an arbitrary Borel measure)? $$ \{x \notin P : d(x, P) = 0\} $$ where $d(x, P) = \inf_{z \in P} \{ d(x, z) \}$ and $d(\cdot, \cdot)$ is the distance function.