Suppose I have a random variable $ X: \Omega \longrightarrow R$. Where $\Omega$ is the sample space.
Suppose then that I have another random variable $Y$ which is some function of $X$, $Y=f(X)$.
What is the sample space of $Y$? Is it $\Omega$ or $R$?
If the sample space is still $\Omega$ can I also say that $A_1: X<0 , A_2: X \geq 0$ is a partition of $\Omega$ ?, or is it only a partition of $R$ since $X$ is real valued?
I am asking this question because I want to apply the law of total expectations on $Y$, i.e. I want to express $E[Y]$ as
$E[Y]=E[Y|X<0]P(X<0) + E[Y|X \geq 0]P(X \geq 0)$