Writing a r.v. in set notation

[nasshi]

The definition I have for a random variable is
X=\lbrace \omega \in \Omega \vert X(\omega) \in B \rbrace \in F where F is a sigma algebra and B is a Borel subset of R.

Using function composition, how would one write a similar set notation definition for f(X), where f is a Borel measurable function?
f(X)=\lbrace \omega \in \Omega \vert f(X(\omega)) \in B \rbrace \in \sigma(X) where \sigma(X) is a sigma algebra and B is a Borel subset of R??

[mathman]

Your espression looks right. In both cases, there is an implicit assumption that X and f(X) are real-valued.