# Size 1/2 Borel set homogeneously scattered across the unit interval [closed]

I’m looking for a Borel subset A of the unit interval with the property that, for any subinterval J of the unit interval, the measure of the intersection of A and J is one-half the length of J. So giving a decimal expansion of a number x to an large number of decimal places won’t give you any information at all about whether x is in A.

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• en.wikipedia.org/wiki/Lebesgue%27s_density_theorem Commented Sep 4 at 4:06
• Vann, welcome to MathOverflow. Your question is answered negatively by the Lebesgue density theorem, which shows that it is impossible, since every positive measure set must often have regions where the density of the set goes to 1, and indeed, the density is 1 at almost every point in the set. I'm sorry that your question has had a negative reaction, but I would look forward to further contributions from you, since I believe you would have a lot to offer here. Commented Sep 4 at 14:33