Here is one answer, which may answer the question as I understand it (but perhaps I do not understand the question correctly).
Start with a measure $\mu$ defined on a $\sigma$-algebra $\mathcal A$. Then define the outer measure $\mu^*$ associated with $\mu$, in the sense of Caratheodory. Once you have this outer measure, you can define the class $\mathcal A(\mu^*)$ of all $\mu^*$-measurable sets (still in the sense of Caratheodory).
Then, it is part of the Caratheodory extension theorem that $\mathcal A\subseteq \mathcal A(\mu^*)$. So, in this sense, the answer to the question is "Yes".
On the other hand, it is not necessarily tru that all measurable sets belong to the original $\sigma$-algebra $\mathcal A$...