What is the purpose of random variable functions within probability theory? This is a rewrite of the question. See the original here.

For a given probability field $(\Omega, A, P)$, a random variable defined on that field would be this (I'll use a real-valued random variable for simplicity):
$$X : \Omega \mapsto \mathbb{R}$$

As I understand it from the comments, this should be interpreted as this: having observed a random outcome $\omega \in \Omega$, what are the consequences of that outcome?

As an example given by Dilip Sarwate, if I toss a coin and heads comes up, I win \$1.00; if tails comes up, I lose \$0.50. In that case, the random variable would be $X(H) = 1.00,\ X(T) = -0.50$.

If I got all of the above right, here's my question. Random variables (ie. consequences of a random outcome) seem like a very high-level concept. Why are they necessary within an abstract mathematical theory such as probability theory?
Kolmogoroff's probability axioms and the notion of random variable are a prime paradigm of $20^{\rm th}$ century mathematics. To compute probabilities connected with coins and urns you maybe could do without random variables; but if you want to go at complicated stochastic phenomena like the weather you absolutely need them.

A random variable has a priori nothing "random" about it: It is a well defined function on a maybe huge "probability space" $\Omega$. An individual point $\omega\in\Omega$ may be the possible "world weather during  24 consecutive hours" and entail information about temperature, clouds, humidity etc. at all points of the earth at all times of a day. Contrasting this ocean of possibilities a real valued random variable $T$ could be the temperature at Kennedy Airport, New York, at 12.00 p.m., on a given day. Given $\omega$, the value of $T$ is well defined, but "chance" or "fate" chooses the point $\omega$ where $T$ is evaluated. Note that it is absolutely impossible to "observe" the point $\omega\in\Omega$ in its totality, but we can observe $T$ on any given day, and we are even able to observe the function $t\mapsto T(t)$.

Kolmogoroff's axioms allow to talk coherently about the "probability that it rains on three consecutive days at Kennedy airport" or about the probability that the temperature is $\leq 31^\circ$ Celsius there at $09.00$ a.m. tomorrow, without really dealing with the intricacies of the space $\Omega$.
I think that we need a random variable just because it is a number. The thing which we all have experience to deal with from the moment we start counting change at a grocery store.