Inclusion of sigma algebra for r.v.'s $X \leq Y$

Let say we have two real-valued random variables satisfying $X \leq Y$ defined on the same probability space. Can we say something (an inclusion relation) concerning the sigma algebras $\sigma (X)$ and $\sigma (Y)$ ?

I am tempted to use the fact that the sigma algebra generated by a r.v. is the littlest one containing all the set levels $\lbrace \omega : X(\omega) \leq \alpha\rbrace$ for every $\alpha \in \mathbb R$ but I'm not sure.

Could you help me with that?

• Nope. That $\sigma(X)\subseteq\sigma(Y)$, say, refers to the fact that $X=u(Y)$ for some measurable $u$. Note that $\sigma(X+42)=\sigma(X)=\sigma(X-42)$. – Did Oct 7 '15 at 20:31
• @Did, put this as an answer? – zhoraster Oct 7 '15 at 20:58