What we defined: Suppose we have a (discrete) probability model $\left(\Omega,P\right)$, where $P$ is the probability function (at least, that was the way it was introduced in a course I took; that means only that $\Omega$ is at most countable, that $P\left(\bigcup_{i}A_{i}\right)=\sum P\left(A_{i}\right)$, for all (at most countable and disjoint) events $A_{i}$ and that of course $P\left(\Omega\right)=1$). We defined a random variable (rd from now on) $X$ to be a mapping $X:\Omega\rightarrow\mathbb{R}$ and then discussed some aspects of $P\left(X=k\right)$ for some $k\in\mathbb{R}$ .
What bothers me: To me, this definition seems rather artificial: Why define a mapping like that, if there is no apparent need for it (at least in this probability model) ?
Since if we have an event $A\subseteq\Omega$, that depends on some parameter $k\in\mathbb{R}$, (for example the sum of the faces of dice be $k$) then we could just as easily define a collection of events $A_{k}$ - one for each parameter - and discuss aspects of $P\left(A_{k}\right)$, instead of the above way by using $X$. Of course one could now argue, that $A$ does not always need to depend on some parameter $k$, so one in some cases really has to "convert" probabilities to numbers via $X$, but of all examples that I have seen until now, even in $\Omega$ isn't made up out of numbers, somewhere a parameter $k$ does sneak in, so we could equivalently work with $P\left(A_{k}\right)$ instead of $P\left(X=k\right)$, since defining the subsets of $\Omega$ whose elements we want to count (to establish the probability of the subset) always amounts to using some $k$ in the definition of those subsets (this reasoning extends of course also to other cases like when we consider $P\left(X\leq k\right)$, since this can also be circumvented by considering an appropriate $P\left(\cup_{j\leq k}A_{j}\right)$).
Thus, introducing rd's seems to me to be a superfluous definition of events without specifying $k$"