How to Define f(X) 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.