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Is there a subset $A\subset\mathbb R$ such that for any interval $I$ of length $a$ the set $A\cap I$ has lebesgue measure $a/2$?

Can it be constructed explicitly?

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You can use Lebesgue's density theorem. If such a set exists then for every point of $A$ the Lebesgue density would be $\frac{1}{2}$. Since, by Lebesgue's theorem the Lebesgue density must be $0$ or $1$ almost everywhere this is a contradiction.

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  • $\begingroup$ Would it be possible to have a type of measure for which this would hold? $\endgroup$ – Sambo Jul 20 '17 at 17:36

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