Inverse function of borel sets when function is a constant.

Question

Following a simple proof my professor explained in class I am having problems with a specific step:

The proof is of probabilistic nature and we are trying to prove that If $X$ (random variable) is independent from $g$ (sigma algebra) then $ E[X|g] = E[X] $. A standard result.

My professor starts by setting $Z = E[X] $ , then he wrote $ Z^{-1}(H) \in \{ \emptyset, \Omega \} \ \forall H \in B $ where $B$ is the Borel set.

But here I am having problems, following the def of inverse function i get: $ Z^{-1}(H) = \{ w \in \Omega | Z(w) \in H \} $ It seems to me that more elements of the sample space could be included in this set other than the whole sample space and the empty set. I would need to know wich elements of the sample space map to $E[X] $, but there could be some $w$ that verifies $ Z(w) \in H $.

What am I misunderstanding?

Answer 1

$Z=E[X]$ is a constant function on $\Omega$.

Answer 2

Suppose that $Z(\omega) = c$ for every $\omega\in\Omega$. Then $Z^{-1}(B) = \Omega$ iff $c \in B$, and $Z^{-1}(B) = \emptyset$ iff $c \notin B$.