Given a Lebesgue measurable set A with strictly positive measure, can we find an open interval (a,b) such that x belongs to A for almost every x in (a,b)?
Thanks in advance for any comments!
Given a Lebesgue measurable set A with strictly positive measure, can we find an open interval (a,b) such that x belongs to A for almost every x in (a,b)? Thanks in advance for any comments! 


The usual Cantor set constructed by removing 1/3 at each step is nowhere dense but has measure 0. However, there exist nowhere dense sets which have positive measure. The trick is to try to remove less, for instance you remove 1/4 from each side of [0,1] during the first step then 1/16 from each pieces etc... The resulting set is the fat Cantor set: it is nowhere dense and it has positive measure. 


I actually worked this problem out during my Measure Theory course a couple years ago. http://www.austinmohr.com/Work_files/hw2_3.pdf Therein, you'll find a construction of a Cantorlike set having any measure strictly between 0 and 1. As with the Cantor set, you cannot find an interval contained in this Cantorlike set. I don't suggest you accept my solution without checking it yourself, as it was written by one still coming to grips with the material. It should give you a good idea of how to construct your proof, however. 


From the positive side, as I've mentioned on the comments, you should look at Lebesgue's density theorem. It says that you will get intervals where the measure of the set intersected with the interval is as close to full as you like, in fact, this can be done for intervals around almost every point of the set, and it holds in other contexts also. 

