From Royden &amp; Fitzpatrick's Real Analysis, Problem 19 section 2.4:  "Let E have finite outer measure. Show that if E is not measurable, then there is an open set O containing E that has finite outer measure and for which  m*(O/E) > m*(O) - m*(E)."  My process goes like this:   1) m(E) is finite => E is bounded => There exists open set O containing E such that for all e1>0, m(O)-m(E)  2) E not measurable => there exists some e2>0 such that there exists an open set Q containing E and m(Q/E)>e2.   If I could somehow show that for the same e>0, Q=O then I would have the property immediately. But I cannot see any way that I can justify equating O with Q, and I do not see any other angle of approaching the problem, anyone can offer any hints or answers, I'd much appreciate it.
(1)  If $E$ is a subset of $\mathbb R$ with finite outer measure, i.e. $m^{*}(E) &lt;\infty$; and   (2) $E$ is not Lebesgue measurable, i.e. there exists $F$ such that  $m^{*} (F) &lt; m^{*}(EF) + m^{*}(FE^{c}).$  [Claim] There exists an open set $O\supset E$ with finite outer measure, such  that   $$m^{*}(O E^{c})  &gt; m^{*} (O)  - m^{*} (E).$$  My  questions are the following.  (1) Is the above claim correct? I've seen this in Royden's book, but have a bit concern. Note that a set $E$ is Leb measurable if  it satisfies caratheodory condition: $m^{*} (F) = m^{*}(EF) + m^{*}(FE^{c})$ for  all subset $F$. If the claim is yes, it seems that measurability condition can  be reduced from all subset $F$ to Borel set $F$.  (2) If the claim is correct, a proof is needed.
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What EXACTLY cause Voldemort's soul piece to split off when he killed Lily and tried to kill Harry?