TITLE: Why do we need (the abstract concept of) random variables (in discrete probability models)?
QUESTION [18 upvotes]: 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 $k$"

REPLY [5 votes]: In principle, you can get away with joint distribution laws instead of random variables but the things get extremely messy and formulations ridiculously tangled (try to reformulate the strong law of large numbers this way and you'll see what I mean). In addition, you lose all intuition coming from classical analysis and the Lebesgue integration theory. Even such a simple operation as the truncation of a random variable at some level will become the projection of the distribution measure, etc. So, why to have a headache expressing everything in a fancy way when a nice and powerful language is available? 
Of course, the Kolmogorov formalism may lead to some confusion too. For instance, if you have a random variable on $\Omega$, and want to introduce an independent copy (a standard trick in many proofs), it may be just impossible to do it on the same $\Omega$, so, technically speaking, a random variable is not quite the same as a function on a measure space; you should rather say that it can be realized as a function on a measure space and to check that the properties you define do not depend on the realization and are determined by the (joint) distribution only. Still, after proving it three times or so, it becomes fairly obvious what is legitimate and what is not and you can safely forget about fine issues like this one (There are more of them; the next one comes when you need conditional probabilities in the continuous setting, which requires the underlying $\Omega$ to be a Polish space, etc.).