I have proved the follow:
Let $X$ be a set.
Let $S$ be a semi-ring of subsets of $X$.
Let $\mu$ be a premeasure on $S$.
Let $\overline{\mu}$ be a premeasure on a ring generated by $S$ which extends $\mu$.
Then the outer measures induced by each are identical.
So a natural question arose here:
Question
Let $X$ be a set
Let $R$ be a ring of subsets of $X$.
Let $\mu$ be a premeasure on $R$.
Let $\Sigma$ be the $\sigma$-algebra generated by $R$.
Let $\overline{\mu}$ be a measure on $\Sigma$ which extends $\mu$
Are the outer measures induced by each the same? If not, is it true when $\mu$ is $\sigma$-finite?
I guess this is not true since $\Sigma$ cannot be reached by $R_{\sigma \delta ...}$ (countable times), but i cannot think of a counterexample