Halmos developed measure theory based on $\sigma$-rings. Nowadays measure theory is based on $\sigma$-algebras. I would like to know how to bridge the two theories if possible.
Namely, let $(X, \mathcal R, \mu)$ be a measure space, where $\mathcal R$ is a $\sigma$-ring. Let $\mathcal A$ be the smallest $\sigma$-algebra containing $\mathcal R$. Can $\mu$ be extended to a measure on $\mathcal A$?. If yes, is the extension unique?
My guess is that if $\mu$ is $\sigma$-finite, i.e. every member of $\mathcal R$ is covered by a countable union of members of finite measure, then the extension is unique.