User ehsan - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T06:19:19Z http://mathoverflow.net/feeds/user/2619 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8972/do-sets-with-positive-lebesgue-measure-have-same-cardinality-as-r/9088#9088 Answer by Ehsan for Do sets with positive Lebesgue measure have same cardinality as R? Ehsan 2009-12-16T07:16:15Z 2009-12-16T07:16:15Z <p>I am not sure if this answer can be helpful or not, but since it is a very elemantary approach to the problem, it might be useful. Assume we are given a subset of R such as A with cardinality smaller than R. Then you can show that the cardinality of A and A+A for any infinite set A is the same. Hence the cardinality of A+A is the same as the cardinality of A which is smaller than R. But you can prove that for any set A of positive measure, A+A has at lease one open interval as a subset. This will be a contradiction with the fact that cardinality of A+A is smaller than R.</p> http://mathoverflow.net/questions/8972/do-sets-with-positive-lebesgue-measure-have-same-cardinality-as-r/9088#9088 Comment by Ehsan Ehsan 2009-12-19T01:13:02Z 2009-12-19T01:13:02Z 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. http://mathoverflow.net/questions/8972/do-sets-with-positive-lebesgue-measure-have-same-cardinality-as-r/9088#9088 Comment by Ehsan Ehsan 2009-12-16T16:29:05Z 2009-12-16T16:29:05Z 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. http://mathoverflow.net/questions/8972/do-sets-with-positive-lebesgue-measure-have-same-cardinality-as-r/9088#9088 Comment by Ehsan Ehsan 2009-12-16T16:16:22Z 2009-12-16T16:16:22Z Let's prove that if A is of positive measure then A+A contains an interval. First show that if m(A)&gt;0 then there is an open interval L such that m(A intersect L)&gt;(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.