How to find outer maesure of difference of two sets?
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The Lebesgue measure and Lebesgue outer measure coincide on Lebesgue measurable sets, which can be defined in several equivalent ways. Let mm and m∗m∗denote the Lebesgue measure and the Lebesgue outer measure respectively. These are some possible definitions of A⊂RnA⊂Rn being measurable:
For all B⊂RnB⊂Rn
m∗(B)=m∗(B∩A)+m∗(B∖A)m∗(B)=m∗(B∩A)+m∗(B∖A)
For all ϵ>0ϵ>0 there exist an open set GG and a closed set FF such that F⊂A⊂GF⊂A⊂G and m∗(G∖F)<ϵm∗(G∖F)<ϵ. (Note that since G∖FG∖F is open, it is measurable, so that m∗(G∖F)=m(G∖F)m∗(G∖F)=m(G∖F).)
A=F∪NA=F∪N, where FF is an FσFσ (i.e. a countable union of closed sets) and m(N)=0m(N)=0.
A=G∖NA=G∖N, where GG is a GδGδ (i.e. a countable intersection of open sets) and m(N)=0m(N)=0.
The reason for the need of two different concepts is that neither of them is "perfect":
mm is a measure, but is not defined for all subsets of RnRn
m∗m∗ is defined for all subsets of RnRn, but is not additive: here exist disjoint sets AA and BB such that m∗(A∪B)≠m∗(A)+m∗(B)m∗(A∪B)≠m∗(A)+m∗(B).
For all B⊂RnB⊂Rn
m∗(B)=m∗(B∩A)+m∗(B∖A)m∗(B)=m∗(B∩A)+m∗(B∖A)
For all ϵ>0ϵ>0 there exist an open set GG and a closed set FF such that F⊂A⊂GF⊂A⊂G and m∗(G∖F)<ϵm∗(G∖F)<ϵ. (Note that since G∖FG∖F is open, it is measurable, so that m∗(G∖F)=m(G∖F)m∗(G∖F)=m(G∖F).)
A=F∪NA=F∪N, where FF is an FσFσ (i.e. a countable union of closed sets) and m(N)=0m(N)=0.
A=G∖NA=G∖N, where GG is a GδGδ (i.e. a countable intersection of open sets) and m(N)=0m(N)=0.
The reason for the need of two different concepts is that neither of them is "perfect":
mm is a measure, but is not defined for all subsets of RnRn
m∗m∗ is defined for all subsets of RnRn, but is not additive: here exist disjoint sets AA and BB such that m∗(A∪B)≠m∗(A)+m∗(B)m∗(A∪B)≠m∗(A)+m∗(B).
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