Let $\mu$ be a measure defined on a certain $\sigma$-algebra $\Sigma$, and let $R,Q\subseteq \Sigma$ be two different semi-rings of sets within it, such that their generated $\sigma$-algebras coincide. That is: $$\sigma(R)=\sigma(Q)$$
If we restrict $\mu$ to these semi-rings we will get pre-measures on them, which we can then extend via Caratheodory. Will the two $\sigma$-algebras of Caratheodory measurable sets generated by these pre-measures necessarily be the same $\sigma$-algebra?
Phrased another way, can we take a pre-measure, extend it, then restrict it to a new but "just as large" semi-ring such that after re-extending it we'll end up with different measurable sets?