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Jun
25 |
awarded | Yearling |
Mar
25 |
awarded | Good Answer |
Mar
6 |
awarded | Nice Answer |
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? |