# An outer measure is built from a collection of sets, then are these sets measurable?

We know

Let $\epsilon\subset\mathscr{P}(X)$ and $\rho:\epsilon\rightarrow[0,\infty]$ be such that $\emptyset\in\epsilon, X\in\epsilon,\rho(\emptyset)=0$, then the set function defined on $\mathscr{P}(X)$: $$\mu^*(A):=\inf\{\sum_1^\infty\mu(E_j):E_j\in\epsilon\ and\ A\subset\bigcup_1^\infty E_j\}$$ is an outer measure.

My question is: are sets in $\epsilon$ always $\mu^*-$measurable? If not, could anyone give some examples?

• Presumably in the def'n of $\mu^*(A)$ you mean to write $\rho (E_j)$, not $\mu (E_j).$ Note that then $\mu^*(A)$ is defined for every $A\subset X$. But what is your def'n of "measurable"? – DanielWainfleet Dec 12 '17 at 9:01

No. One-dimensional Hausdorff measure in the real line (= Lebesgue measure in the real line) may be defined using all subsets with $\rho(A)$ the diameter of $A$. But of course some sets are not Lebesgue measurable.