# Describing the outer measure generated by a set function.

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.

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There aren't that many countable collections from $\mathcal{F}$, are there? What can you cover with those? – Henno Brandsma Feb 11 '12 at 6:09
I can can cover $\{1\},\{2\},\{3\},(1,2), (1,3)$ and $[1,3]$. Is that right? – Bill Feb 11 '12 at 6:13
Surely you can imagine more subsets you can cover? What about $\mathbb{Q} \cap [1, \frac{3}{2}]$, to name but one? You seem to realize you cannot cover any sets that have points outside $[1,3]$, so what sets are those? – Henno Brandsma Feb 11 '12 at 6:41