This was an exam question. I know that my answer is wrong, but I believe myself to be on the right track. Can someone help me finish my construction? Here is the question.
Find a set $E$ with the property that for any open interval $(a,b)$, $m(E \cap(a,b))= \frac {b-a}{2}$ or prove such a set cannot exist.
Here's what I figure so far:
1) Obviously $E$ itself is not an interval. If it was, $m(E \cap E)=m(E)/2,$ which is only true if $m(E)=0,$ so $E$ is not an interval.
2) Furthermore, $E$ does not contain any intervals. If it did, then choose $E' \subset E,$ where $E'$ is an interval. Then $m(E' \cap E)=m(E')=m(E')/2$ as before. So $E'$ is not an interval and therefore $E$ contains no intervals.
Now, during the exam, my brain jumped to a Cantor set, so I attempted to build a Fat Cantor Set based on removing $2^{k-1}$ pieces of length $\frac {1}{2^{2k}}$ from the middle of each unit interval in $\Bbb R$. I know that this set has measure $1/2$, but if $(a,b)$ is one of the intervals deleted in the construction of the cantor set, the measure of the intersection is not half the measure of the interval. There are also infinitely many unit intervals in $\Bbb R,$ so in order to have disjointness, I'd have to be tiling $\Bbb R$ with these sets, but then clearly taking any set not centered appropriately doesn't meet the hypothesis either. So this set doesn't work as I thought it did.
In hindsight, I'm skeptical as to whether such a set actually exists. But how to prove it?
Edit: A particularly nice solution would be one that gets me from $E$ has no intervals to $E$ does not exist.
