Assume that you have a probabilityts pace $(\Omega, \mathcal{F},P)$, and two random variables X, Y $(\Omega, \mathcal{F})\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Let $\sigma(X,Y)$, be the sigma algebra on $\Omega$ generated by all the sets of the form $X^{-1}(B), Y^{-1}(B)$. And let $\sigma(X-Y)$, be the collection of sets $\{(X-Y)^{-1}(B)\}$.
Does it then hold that $\sigma(X-Y)\subset \sigma(X,Y)$?
It seems very plausible that this is the case, and I need it for another result I have to prove. But I am not able to finish this proof.
My only idea is the smallest sigma algebra trick. That is, prove that the sets having the property is a sigma-algebra, and that it contains the generating sets. Proving that the sets having this property is a sigma algebra is fairly easy since $X-Y=Z, Z^{-1}(\emptyset)=\emptyset, Z^{-1}(B)^c=Z^{-1}(B^c), Z^{-1}(\cup B_n)=\cup Z^{-1}(B_n)$.
But this only reduces the problem a little, then I still need that for any open set O, $(X-Y)^{-1}(O)\subset\sigma(X,Y)$. So this does not seem like a good strategy.(I could reduce it further, using that the rationals are dense, and then I only need the result for open intervals, but I do not see how that solves it either.)
Any hints or help? Or is what I am trying to prove not true? It may be that it fails, but I doubt it.
Best Answer
The $\mathbb{R}^2$-valued random variable $(X,Y)$ is measurable wrt. $\sigma(X,Y)$ and so is any continuous function of $(X,Y)$. Thus $X-Y$ is measurable wrt. $\sigma(X,Y)$, that is, $\sigma(X-Y) \subset \sigma(X,Y)$.