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?
 A: 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 $\Omega$.
A: The axioms of probability theory were designed to fit with the existing "layman's" notion of probability, i.e. the chances (a real number), of an certain event ($\omega$) occurring, from a set of given possible outcomes ($\Omega$).
In probability theory, like in most branches of mathematics - necessity is the mother of invention, not the other way round.
A: Probability theory grew out of the mathematical treatment of gambling. It came to live as a very concrete subject. There are several ways to formalize probability and it is possible to formulate probability in a completely point-free way. So you dispose of the space $\Omega$ and just work with a Boolean $\sigma$-algebra of events and a probability measure defined on it. Random variables can then be defined as Boolean $\sigma$-homomorphisms. If you want to recover the points, you can take $\Omega$ to be the space of maximal consisten descriptions in terms of events (they are ultrafilters). This approach is well described and motivated in the beautiful paper On the axiomatic treatment of probability (1955) by Jerzy Łoś. The cost is an additional layer of abstraction.
