This is a follow up on: Is a random variable constant iff it is trivial sigma-algebra-measurable?. I wanted to comment on the solution, but, unfortunately, this is only possible from 50 reputation.
I just started learning probability theory and am struggling with the following proof:
Let $(\Omega, F, P)$ be some probability space and $X$ a random variable on that space. Prove the following: If $F$ is the trivial sigma algebra then $X$ must be a constant.
Adapted from the link above (this is only the first part of the proof for the converse refer to the link):
Suppose $F=\{\emptyset,\Omega\}$, and there exists $a,b \in \Omega$ such that $X(a) \neq X(b)$. Then $X^{-1}(\{X(a)\}) \notin F$, is a contradiction. It follows that $X$ is constant.
My question: Does it not have to be $X^{-1}(\{X(a)\})$ or $X^{-1}(\{X(b)\}) \notin F$? Since one of them might still be element of $F$, no?
Thanks for any help!
Cheers, dahlai
