There seem to be two general approaches to the development of Random Variables, enforcing a choice between ``feast'' or ``famine'' of rigor. The first is to simply say a random variable is a variable which takes on values according to some probability distribution, and leave it at that. The second is to delve into a subtle realm of measure and probability theory, rich and involving fields unto themselves. We will attempt to steer a middle course between this Scilla and Charybdis, using what we need of the indubitably elegant foundation laid by Kolmogorov and others without sinking too far into chthonian abstraction.
Let us begin with a state space. This is a finite dimensioned Hilbert Space, generally 
.
We will assume the concept of a Random Variable, with finite variance, taking on values in this
space (see Appendix). Consider further the infinitude of possible such Random Variables (RV's). It
is the relationships between these that determine the stochastic behavior of our system, and
it is on the set of all such RV's that we shall build our edifice. For this set may be shown
to be a Hilbert Space itself.
This will be of great
interest to us, because it allows us to treat probabalistic relationships as geometric ones.
This is of great mathematical, computational, and aesthetic import.
We may develop this vector space informally by merely noting that the operations of vector addition and scalar
multiplication defined on the original state space naturally induce definitions for the space of Random
Variables. Linear combinations of RV's may be evaluated in the obvious way, yielding still more elements of 
,
our space of Random variables.
The closure of the original set under these operations ensures the closure of the
space of random variables, and similarly with the rest of the Vector Space Axioms.
To proceed slightly more formally, we begin by developing the one dimensional case. Consider the set
of all finite variance real-valued random variables. We may associate this with the set of all
square-integrable real-valued functions, 
. We will consider to be simply the
Cartesian Product of this infinite
dimensional space of scalar random variables with itself d times.
It is the similarity between this construction and that of which provides its
utility and intuitive advantages, and it is the differences which make for a rich
landscape of mathematical structures and challenges.