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Is there a measurable subset E ⊆ R such that whenever a < b are real numbers we have both $m(E ∩ (a, b)) > 0$ and $m((a, b) -E) > 0$ ? This is an extra question on my real analysis class, but it is not graded, just for fun.
I was thinking about the "fat Cantor set"(https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set), and play with it a little bit, but somehow I can still find "gaps" between real numbers. Can anyone give me some hint so I can carry on ? Thanks !!!