Prove for any set $E\subset R$ with Lebesgue measure 1 there exists a subset with Lebesgue measure 1/2. It looks easy but I have tried for an hour and could not find a way to prove it. Can anyone provide a hint? Like what theorem or technique to use? Thank you!
As hj implied, this is true for any value in $[0,1]$. I remember this from one of the Rudin books. A similar approach to his/her post, for any value $\alpha$ in between (for Lebesgue measure): consider a compact subset $K$ with $m(K)> \alpha $. Then $K$ is contained in a interval $I=[c,d]$ , then use the function $M(x):m([c,x] \cap I )$, and the intermediate value theorem, that we can use, as hj said, by continuity.