I have maybe a naive question about compactness of outer measure (or completion).
Let $(E,\mathcal B(E))$ be a Polish space, and $\mathcal M_b(E)$ the bounded Radon measure on $E$. Assume that a sequence $\{\mu_n\}$ weakly converge to $\mu$ in $\mathcal M_b(E)$ i.e. $$ \int_E \varphi \mu_n \to \int_E \varphi \mu \,, \quad \forall \varphi\in \mathcal C_b(E,\mathbb R)$$ where integration (and continuity) is made throuth the Borel $\sigma$-algebra $\mathcal B(E)$.
Is it true that the same holds for the completion of $\mu_n$? Namely, let $\mu_n^*$ the (unique) completion of $\mu_n$ for all $n$, do we have $$ \int_E \varphi \mu_n^* \to \int_E \varphi \mu^* \,,\quad \forall \varphi\in \mathcal C_b(E,\mathbb R)$$ where integration (and continuity) is made throuth the extended $\sigma$-algebra and $\mu^*$ the (unique) completion of $\mu$.
I do not realize if it is trivial or not... Maybe some hints?