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Let $A$ be a subset of $\Bbb R$ with $\lambda^*(A)>0$. Show that there exists a nonmeasurable subset $B$ of $\Bbb R$ s.t. $B$ is a subset of $A$

I'm a little confused where to start with this one. If $A$ is nonmeasurable there is nothing to prove so we have to assume that $A$ is measurable, more specifically a nonzero measure. Define an equivalence class next?? not sure....

Brian M. Scott
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cele
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    What exactly is $\lambda^*(A)$ here? – aschepler Dec 09 '13 at 01:30
  • @aschepler: Presumably Lebesgue outer measure. – Brian M. Scott Dec 09 '13 at 01:31
  • yes, @aschepler, it means the outer measure generated by the Lebesgue measure on R – cele Dec 09 '13 at 01:56
  • Have you seen the construction of a Vitali set? – Brian M. Scott Dec 09 '13 at 01:58
  • @BrianM.Scott have seen it in text but have not done too much work using it. my thinking here is use the Axiom of choice to show there exists a subset B of A containing precisely one member from each equivalence class. let {r1, r2, ....} be an enumeration of the rationals [-1,1] and let Bn=rn+B. Im still working on how to show this and finish from here – cele Dec 09 '13 at 02:03
  • You can’t guarantee that each equivalence class meets $A$, but you can use the axiom of choice to say that there is a $B\subseteq A$ such that if an equivalence class actually intersects $A$, then $B$ contains exactly one member of that class. However, before you do that you should replace $A$ by $A\cap[n,n+1]$ for some $n$ such that $\lambda^*(A\cap[n,n+1])>0$, so that you’re working with a bounded set. – Brian M. Scott Dec 09 '13 at 02:13
  • what if, following from my previous comment, i stated that the sequence {Bn} is pairwise disjoint, lamda(Bn)=lamda(B) holds for each n and that A is a subset of the union from 1 to infinity of Bn which is a subset of [-1,2]. Noting that if B is a measurable set, then each Bn is measurable set, proof by contradiction? @BrianM.Scott – cele Dec 09 '13 at 02:24
  • Not a subset of $[-1,2]$, but rather of $[n-1,n+2]$, but otherwise you’re okay: $A$ has positive measure, so the sets $B_n$ cannot be null sets, but then their union can’t be a subset of the finite interval $[n-1,n+2]$. – Brian M. Scott Dec 09 '13 at 02:27
  • Look here: http://math.stackexchange.com/questions/206618/positive-outer-measure-set-and-nonmeasurable-subset –  Dec 09 '13 at 14:53

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