Let $\mu$ be a Borel outer measure on $\mathbb{R}$, and suppose that for any open interval $I$, we have $\mu(I) = l(I)$, the length of $I$. Then $\mu(B) = \mathscr{L}(B)$ (a Lebesgue outer measure) for all Borel sets $B$.
I think that the conclusion holds since $\mu$ is an outer measure restricted to Borel $\sigma$-algebra and $\mathscr{L}$ is an outer measure restricted to $\sigma$-algebra and Borel $\sigma$-algebra is a subset of $\sigma$-algebra. Therefore, these are the same for all Borel sets. Is it correct?
If yes, why do we need the hypothesis that $\mu(I) = l(I)$?
Could you explain what I am missing here?
Thank you in advance.