A collection of sets $O_{\lambda}$ open covers a set $A$ in $\bf R$ that is bounded and measurable. Assume that $A$ can not be open covered by a finite subcollection of $O_{\lambda}$, let $\epsilon>0$. Is there a finite subcollection of $O_{\lambda}$, denote $O_n$ (don’t have to be disjoint), such that $m^*(A\setminus\bigcup O_n)<\epsilon$?
Also, is it equivalent to say "the subset of $A$ that can not be finitely open covered by some subcollection of $O_{\lambda}$ has measure 0" ? edit: (this paragraph is not mathematically constructed right, attribute to Wong pointing it out, but would happy to take other good alternate rephrases)
