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This question tortures me for a while: if a set $E$ has positive Lebesgue measure, does it necessarily contain an interval? I would be truly grateful for help.

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No, think about the Smith–Volterra–Cantor set. – MerylStreep Feb 14 at 4:16
    
Thank you. I will look it up. – Marina Feb 14 at 4:18
up vote 8 down vote accepted

The irrational numbers are such an example.

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+1. Small comment: it's possible to come away from this problem, and facts like regularity, with the impression that a positive measure set must almost contain an interval: that is, for all $E$ measurable with positive measure, there is a nontrivial interval $I$ such that $I\subseteq E\cup Z$ for some measure-zero set $Z$. However, this is false: see math.stackexchange.com/questions/57317/…. – Noah Schweber Feb 14 at 4:24
    
@NoahSchweber Good point; that's a good example of how Littlewood's three principles can sometimes be a bit misleading. – T. Bongers Feb 14 at 4:25
    
I feel totally mislead. With the "fat" Cantor set, for example, they say that it is the boundary of the set which brings that positive measure, not the set itself. But since the set is closed, is not the boundary included into the set? – Marina Feb 14 at 4:30
    
@Marina Yes, that set is closed and is equal to its own boundary. – T. Bongers Feb 14 at 4:32
    
Things like this make me believe that the real numbers are not continuous. – Marina Feb 14 at 4:34

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