Can an atom of a measure be reduced to a single point in a complete separable metric space?

Question

Let $X$ be a complete separable metric space and $\mu$ a Borel measure that is boundedly finite, i.e. $\mu(B) < \infty$ for all bounded Borel sets $B \subset X$. Call a Borel set $A$ an atom if $\mu(A) > 0$ and $$ B \subset A, \quad \mu(B) > 0 $$ imply $\mu(B) = \mu(A)$. If $A$ is an atom, does there exist a point $x \in A$ such that $\mu$ is concentrated on $x$ in $A$, i.e. $\mu(A) = \mu(\{x\})$? I believe I have a proof of this but can't find this or a similar result being stated anywhere. Thanks for any help.