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Jun
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awarded  Yearling
Mar
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awarded  Good Answer
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Dec
19
comment Do sets with positive Lebesgue measure have same cardinality as R?
1-This is one of the fundamental theorems about the measurable sets that you can approximate their measure from above by open sets. You can find a proof in real analysis books like Folland. 2-Yes the proposition will be still true if you consider any positively measurable sets A and B. Thanks for the comment, i just wanted to make it as easy as possible.
Dec
16
comment Do sets with positive Lebesgue measure have same cardinality as R?
Sorry that I proved above that A-A contains an interval. This is also true for A+A but the proof will be slightly different. Anyways we could start with A-A in the proof of the original problem.
Dec
16
comment Do sets with positive Lebesgue measure have same cardinality as R?
Let's prove that if A is of positive measure then A+A contains an interval. First show that if m(A)>0 then there is an open interval L such that m(A intersect L)>(3/4)m(L). Now use this to show that A-A contains the interval K=(-0.5m(A),0.5m(A)). For the last part let b be a number inside K. Consider all the pairs inside L that their subtraction is equal to b. Prove that A contains at least one of those pairs otherwise the inequality at the begining of the proof can not be true.
Dec
16
awarded  Teacher
Dec
16
answered Do sets with positive Lebesgue measure have same cardinality as R?