Is the set of measurable subsets the completion of the generated sigma-algebra?

Question

Let $\mathcal{R}$ be a semiring of subsets of the non-empty set $\Omega$, and let $\mu:\mathcal{R}\rightarrow[0,\infty]$ be countably additive. Denote by $\mu^\star$ the outer measure on $\mathbb{P}\Omega$ induced by $\mu$, and denote by $\mathcal{M}$ the collection of measurable subsets of $\Omega$ determined by $\mu^\star$. Is $\mathcal{M}$ the $\mu$-completion of $\sigma(\mathcal{R})$? What if $\mu$ is $\sigma$-finite?