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I was just doing thinking about some things we recently did in measure and integration and stumbled upon a question I could not answer.

The scenario is in the reals with standard topology and we take the normal Lebesgue measure.

If I have a perfect set with positive measure, e.g. a fat Cantor set, and take an open interval such that the intersection of the two sets is not empty, is it true then that the intersection also has positive measure?

Intuitively the answer seems yes, but neither could I prove it nor find a counterexample. In addition, if the answer is no, is there a reasonable condition that once added would guarantee that the intersection still has positive measure?

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    $\begingroup$ For a counter example, could you take your perfect set of positive measure to be a fat cantor set union a shifted regular cantor set? Then you could intersect an interval with only the regular cantor set and get a non-empty set of measure zero. $\endgroup$ – michaelhowes Apr 17 '18 at 22:27
  • $\begingroup$ As in the first comment, consider the union of a fat Cantor subset of $[0,1]$ with $\{x+1:x\in C\}$ where $C$ is the Cantor set. $\endgroup$ – DanielWainfleet Apr 18 '18 at 1:22

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