On the collection $S=\{\phi , [1,2] \}$ of subsets of $R$, Define the set function $\mu:S\to[0,\infty)$ by $\mu(\phi)=0$ and $\mu([1,2])=1$. Determine the outer measure $\mu^*$ induced by $\mu$ and the $\sigma$-algebra of $\mu^*$-measurable sets.
for this I attempt with the Caratheodory's method.
That is $\mu^*:P(R)\to[0,\infty]$ ($P(R)$ is the power set of $R$).
Defined as:
$\mu^*(\phi)=0$ and
$\mu^*(A)=\inf\{\sum\limits_{k=1}^\infty\mu(E_k):E_k\in S, A\subset\bigcup\limits_{k=1}^\infty E_k\}$
Since there are only two elements in $S$ IF I'm not mistaken, I identified the $\mu^*$ as,
for $\phi\neq A\subset[1,2]$, $\mu^*(A)=1$ and
otherwise $\mu^*(A)=\inf\{\phi\}=\infty$.
But how do I find the sigma algebra of measurable sets.
That is if a set $E$ is measurable under $\mu^*$ then for any $A\subset R$,
$\mu^*(A)=\mu^*(A\cap E)+\mu^*(A\cap E^c)$ should satisfies.
I tried with the brute force method to identify such sets. But it becomes very cumbersome..
Appreciate your help