Let $X$ be a set, $\mathcal{A} \subseteq \mathcal{P}(X)$ be an algebra (i.e., a set closed under complements and finite unions), $\mu_0$ be a premeasure defined on $\mathcal{A}$, and $\mathcal{M}$ be the $\sigma$-algebra generated by $\mathcal{A}$. Denote $\mu^*$ to be the outer measure induced by $\mu_0$; explicitly, for $E \in \mathcal{P}(X)$, define $$\mu^*(E) = \inf \left\{\sum_{i=1}^{\infty}\mu_0(E_i):\ \{E_i\}_{i=1}^{\infty}\subseteq \mathcal{A} \text{ is disjoint and } E \subseteq \bigcup_{i=1}^{\infty}E_i\right\}.$$ Further, define $$\mathcal{M}^*=\left\{E \in \mathcal{P}(X): \forall K \in \mathcal{P}(X), \;\mu^*(K) = \mu^*(K\cap E) + \mu^*(K\cap E^c)\right\}.$$ That $\mathcal{M}^*$ is a $\sigma$-algebra, $\mathcal{A} \subseteq \mathcal{M}^*$, and hence $\mathcal{M} \subseteq \mathcal{M}^*$ are known facts. When does $\mathcal{M} = \mathcal{M}^*$? Since $\mu^* \Big|_{\mathcal{A}} = \mu_0$, $\mu^*\Big|_{\mathcal{M}}$ is a measure that extends $\mu_0$, and $\mu^* \Big|_{\mathcal{M}^*} $ is a complete measure, the above equality would imply $\mu = \mu^*\Big|_{\mathcal{M}} = \mu^*\Big|_{\mathcal{M}^*}$ is a complete measure extending $\mu_0$.
Now, leaving the question of the above equality aside, denote $\mu^{**}$ to be the outer measure induced by $\mu = \mu^*\Big|_{\mathcal{M}}$; that is, for $E \in \mathcal{P}(X)$, define $$\mu^{**}(E) = \inf \left\{\sum_{i=1}^{\infty}\mu(E_i) = \sum_{i=1}^{\infty}\mu^*(E_i):\ \{E_i\}_{i=1}^{\infty}\subseteq \mathcal{M} \text{ is disjoint and } E \subseteq \bigcup_{i=1}^{\infty}E_i\right\}.$$ What is the relationship between $\mu^*$ and $\mu^{**}$? More generally, if I define $\mu^{\overbrace{*\ldots*}^{n+1}}$ to be the outer measure induced by $\mu^{\overbrace{*\ldots*}^{n}}\Big|_{\mathcal{M}}$ inductively for $n \in \mathbb{N}$ as above, what is the relationship between these outer measures? What is the pointwise limit of this sequence of induced outer measures?
Examples elucidating your arguments will be much appreciated.