Completeness of metric space induced by outer measure (similar to Nikodym metric)

Question

Let $S_\mu$ be a semi-ring of subsets of $X$ and $\mu$ be a $\sigma$-additive measure on $S_\mu$. Let $\mu^*$ be the induced outer measure on $P(X)$.
Define a relation $\sim$ on $P(X)$ by \begin{align} A\sim B\iff\mu^*(A\Delta B)=0 \end{align} for any $A$, $B\in P(X)$.
Take $\mathcal{M}:=P(X)/\sim$, a metric $\rho$ on $\mathcal{M}$ is defined by \begin{align} \rho(\tilde{A},\tilde{B})=\mu^*(A\Delta B) \end{align} for any $\tilde{A}$, $\tilde{B}\in\mathcal{M}$, where $A$ and $B$ are their representatives, respectively.

Question: Is $(\mathcal{M},\rho)$ complete?

The metric $\rho$ is similar to Nikodym metric. The difference is that it is induced by outer measure instead of measure, and the $\sigma$-algebra here is $P(X)$ instead of the collection of all measurable sets. Since outer measure is not $\sigma$-additive, the technique to prove the completeness of a Nikodym metric space does not apply here. I am even not sure whether it's complete or not. I appreciate anyone who can solve it.