# Existence of sequence of measurable sets with prescribed densities

Consider Lebesgue measure $$m$$ on $$[0, 1]$$. Fix a countable sequence $$a_i, 0 < a_i < 1$$ such that $$\sum_i a_i = 1$$. Is there a sequence of disjoint measurable subsets of $$[0, 1]$$, $$E_i$$ whose measure in every open interval $$I$$ respectively is $$a_i m(I)$$?

There is no measurable subset $$E$$ of $$[0,1]$$ such that $$m(E\cap I)=m(I)/2$$ for every open interval $$I\subseteq [0,1]$$.
Indeed, assume there is such $$E$$. Then $$m(E)=1/2$$, so there is an open set $$U$$, $$E\subseteq U \subseteq [0,1]$$ such that $$m(U)=3/4$$. But $$U$$ is a union of a sequence of pairwise disjoint open intervals $$I_j$$, $$j=0,1,2,\dots$$. Since $$m(E\cap I_j)=m(I_j)/2$$ for every $$j$$, it follows $$m(E)=m(E\cap U)=m(U)/2=3/8\neq 1/2$$, a contradiction.