A function $X:(\Omega , \mathcal{F}) \rightarrow (\mathbb{R}, \mathcal{B}) $ is called a measurable function(or random variable with probability $P$) if $X^{-1} (B)= \{ \omega \in \Omega : X(\omega) \in B\ \}$ $\in \mathcal{F}$ for all $B \in \mathcal{B}$
Here, $\mathcal{B}$ is a Borel $\sigma$ -algebra and $\mathcal{F}$ is a $\sigma$ - algebra of $\Omega$
I am trying to understand this definition, and I am considering a example of $\Omega =$ {$1,2,3$} with $\sigma$ - algebra = {$\phi, \Omega,$ {$1$},{$2,3$}}. If $\omega =1,$ then $X(1) \in B$ for all $B \in \mathcal{B}$, so $X(1)= \mathbb{R}?$
Am I right? Or I am doing something wrong?