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?

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

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?