Let $(\Omega , \mathcal{F}, P)$ be a probability space, and let $(\mathbb{R}, \mathcal{B})$ be a measurable space, where $\mathcal{B}$ is the smallest sigma-algebra of subsets of $\mathbb{R}$ that contains all the open interval in $\mathbb{R}$ (which means $\mathcal{B}$ is a Borel set). Let $X:\Omega\rightarrow \mathbb{R}$ be a measurable map, meaning $$ X^{-1}(A)= \{\omega\in\Omega : X(\omega)\in A\}\in \mathcal{F},\ \forall A\subseteq \mathcal{B}$$
Then my question is:
(1) Will the codomain of the map $X$ equal to the range of map $X$?
(2) If yes, then why?