I need to write the following set in the form of intersection of two sets, each of which is the product of two Borel sets (in on dimension).
$\{(x,y)\ |\ x < y \ \wedge \ y < k\}$
In other words, I want to find A, B , C and D where
$\{(x,y)\ |\ x < y \ \wedge \ y < k\} = (A \times \ B) \ \cap \ (C \ \times \ D) $
where A, B, C and D are Borel sets on the real line (one dimension).
I can write the set in the form of intersection as $ \{(x,y)\ |\ x < y \ \wedge \ y < k\} = \{(x,y)\ | \ x<y\}\ \cap \ \{(x,y)\ |\ y<k\} $
but I cannot manage to write the new sets (on the right) in the form of the product of two borel sets on the real line (one dimension real line)
It can't be done. It requires that $A\supset (-\infty,k)\subset C,$ otherwise for some $r<k$ we have $(r, (r+k)/2)\not \in A\times B$ or $(r,(r+k)/2)\not \in C\times D.$ Similarly, it requires that $B\supset (-\infty,k)\subset D.$ But then $ (A\times B)\cap (C\times D)\supset (-\infty,k)\times (-\infty,k).$
However $S=\{(x,y):k>y>x\}$ is the union of countably many "Borel rectangles", for example $S=\cup \{(-\infty,q)\times (q,k): k>q\in \Bbb Q\}.$ Or we can observe that $S$ is open in $\Bbb R^2,$ and $B=\{(a,b)\times (c,d): a,b,c,d\in \Bbb Q\}$ is a countable base (basis) for the topology on $\Bbb R^2$, so $S=\cup \{b\in B:b\subset S\}.$