2

I am trying to either find an example of such a set, or prove that no such set exists. I know of examples of dense sets with measure $1/2$ on specific intervals, such as $[0,1]$, but I haven't been able to find any set that satisfies this more general property.

Dominic Wynter
  • 6,159

1 Answers1

7

Notice that such a set would have to have Lebesgue density $1/2$ at all points, which contradicts the Lebesgue density theorem. There's nothing special about $1/2$ - any fixed ratio can't work for all intervals.

As a related follow-up, there's a nice exercise in Rudin about the construction of a set $A$ so that $0 < m(A \cap I) < m(I)$ for all intervals $I$.

  • Thanks! Which Rudin book are you referring to? I have Principles of Mathematical Analysis, but I can't find that exercise in the section on measure theory. – Dominic Wynter Feb 09 '16 at 13:30
  • Real and Complex (also called Papa Rudin), chapter 2, exercise 8. –  Feb 09 '16 at 14:22