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?