Hi,


I want to write a proof that relies on the fact that:


There are Borel Sets $A$ and $B$ contained in $\mathbb{R}$ such that


$A \cap B = \emptyset$ and $\lambda(A \cap (x,y)) = \lambda(B \cap (x,y)) > 0$.


Note that $x < y \in \mathbb{R}$ are arbitrary.


I'm fairly sure this is true, but am having trouble coming up with a construction of such sets and it's driving me up the wall. Can anyone help?



Sets with equal positive measure in every interval
The basic step is to construct a nowhere dense set of positive, and controlled, measure. Then iteratively in each interval where the set is empty, you replace by another such set. See for example a paper by Erdös and Oxtoby, *Partitions of the plane into sets having positive measure in every non-null measurable set*.


**EDIT:** This gives $\lambda (A\cap (x,y))>0$ and $\lambda(B\cap (x,y))>0$ for each interval $(x,y)$. We cannot have equality $\lambda (A\cap (x,y))=\lambda(B\cap (x,y))$ in general (see Mike Hall's answer).


