Measure Theory Specially Carathéodory measure. and the sigma algebra induced by that

Question

Let $X$ be a set, and let $T$ be a $\sigma$-algebra on $X$.

Let $f: T \to [0,1]$ be a measure and let $\mu∗ f$ be the Carathéodory outer measure induced by $f$. Is it always the case that $T = T\mu∗ f$?

I wonder if we could use the following fact: you may freely use the fact that on the real line, there exists a Lebesgue measurable subset which is not Borel.

Answer 1

The answer is no. Let $T$ be a sigma-algebra which is not complete with respect to $f$ (there are plenty of examples, ranging from trivial to $B_{\mathbb{R}}$). The sigma-algebra of Caratheodory measurable sets is complete with respect to $f$.