Conditioning an independent sum on one summand

The sets $X^{-1}(B)$, B real Borel, form a sigma-algebra because preimages preserve complements and countable unions; by definition this is $\sigma (X)$. Thus the given rectangle identity is precisely independence of Y from every event of $\sigma (X)$. By [F1], $E[Y\mid \sigma (X)]=EY$ and $E[X\mid \sigma (X)]=X$. Linearity [F2] gives the stated sum formula.

Take $\mathrm{\Omega }=\{0,1{\}}^2$, each atom of mass $1\mathrm{/}4$, and $X(u,v)=u$, $Y(u,v)=2v$. This is a probability space by [F3]. Each coordinate value has mass $1\mathrm{/}2$ and each pair mass $1\mathrm{/}4=(1\mathrm{/}2)(1\mathrm{/}2)$; adding atom probabilities proves all Borel rectangle identities. Since $EY=1$, the conditional mean is $u+1$. Directly, on the u=0 fibre the values 0,2 average to 1, and on the u=1 fibre the values 1,3 average to 2.