From Jacod / Protter: "Probability Essentials", Springer:
Note that even if the state space (or range space) $T$ is not countable, the image T' of $\Omega$ under $X$ (that is, all points $\{i\}$ in $T$ for which there exists an $\omega\in\Omega$ such that $X(\omega) = i$ ) is either finite or countably infinite.
(where $X$ is a function (random variable) from $\Omega$ into a set $T$)
I do not understand this. If $T$ is the uncountable set $\bf R$ (the real numbers), could the image also be uncountably infinite?