I have the following question : a detailed answer should be great !
Let $\mu$ be non-atomic probability measure on the unit circle $S^1$ . Prove that : there is a number $\delta > 0 $ such that when the Lebesgue measure of an interval I is < $\delta$ then $\mu(I) < 1/3 $.
The above is the actual question. However, it seems like that on the unit circle $S^1$, the non-atomic probability measures are similar to absolutely continuous measure with respect to the Lebesgue measure, is it correct ? If not, why not correct ? Is there an example of a non-atomic probability measure on $S^1$ such that it is NOT absolutely continuous w.r.t. the Lebesgue measure ?