Let $\mathcal{F}=\{\emptyset,[1,3]\}$ be a collection of subsets of $\mathbb{R}$. Consider the set function, $\mu:\mathcal{F}\to [0,\infty)$ defined by $\mu(\emptyset)=0,~~\mu([1,3])=2.$
I need help in describing the outer measure $\mu^\ast$ generated by the set function $\mu$ and the $\sigma$-algebra of all $\mu^\ast$-measurable subsets.
I know the outer measure generated by $\mu$ is given by the set function $\mu^\ast : 2^\mathbb{R}\to [0,\infty]$ where $\mu^\ast(E)=\inf \sum_{n=1}^\infty \mu(E_n)$ where the infimum is taken over all countable collections $\{E_n\}_{n=1}^\infty$ of sets in $\mathcal{F} $ that cover $E\subset \mathbb{R}$.
However, I'm a bit lost in proceeding.