0
$\begingroup$

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.

Improve this question
New contributor
Vann McGee is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
$\endgroup$
2
  • 7
    $\begingroup$ en.wikipedia.org/wiki/Lebesgue%27s_density_theorem $\endgroup$
    – David Gao
    Commented 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 Hamkins
    Commented Sep 4 at 14:33

0

Reset to default