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 onehalf 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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7$\begingroup$ en.wikipedia.org/wiki/Lebesgue%27s_density_theorem $\endgroup$– David GaoCommented Sep 4 at 4:06

1$\begingroup$ 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. $\endgroup$– Joel David HamkinsCommented Sep 4 at 14:33
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