$X$ is separable, then the set of all $\mu$-continuity sets forms an algebra

Question

To prove that if $X$ is separable, then the set of all $\mu$-continuity sets forms an algebra.

If $\mu$ is a probability measure on a metric space $X$, a Borel set $A$ of $X$ is called a $\mu$-continuity set if $∂A$ has $\mu$-measure $0$.

Answer 1

It is easy to show that $\partial (A \cup B) \subset \partial A \cup \partial B$. Of course $\partial A =\partial A^{c}$.