In the algebraic axiomatization of probability theory, one of whose main proponents was Irving Segal, the primary concept is not that of probability of an event, but rather that of a random variable. Probability distributions are determined by assigning an expectation to each random variable. The measurable space and the probability measure arise from the random variables and expectations by means of well-known representation theorems of analysis. One of the important features of the algebraic approach is that apparently infinite-dimensional probability distributions are not harder to formalize than finite-dimensional ones. Algebra is the current mathematics collaboration of the week! Please help improve it to featured article standard. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... Probability theory is the mathematical study of probability. ... Irving Ezra Segal (1918-1998) was a mathematician known for work on theoretical quantum mechanics. ... A random variable is a term used in mathematics and statistics. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In mathematics, a measure is a function that assigns a number, e. ...

Random variables are assumed to have the following properties:

1. complex constants are random variables;
2. the sum of two random variables is a random variable;
3. the product of two random variables is a random variable;
4. addition and multiplication of random variables are both commutative; and
5. there is a notion of conjugation of random variables, satisfying (ab)^* = b^* a^* and a^** = a for all random variables a, b, and coinciding with complex conjugation if a is a constant.

This means that random variables form complex abelian *-algebras. If a = a^*, the random variable a is called "real". Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for one of the square roots of negative one (−1). ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, an abelian group is a commutative group, i. ... In mathematics, a *-algebra is an associative algebra over the field of complex numbers with an antilinear antiautomorphism *:A->A which is an involution. ...

An expectation E on an algebra A of random variables is a normalized, positive linear functional. What this means is that

1. E(k) = k where k is a constant;
2. E(a^* a) ≥ 0 for all random variables a;
3. E(a + b) = E(a) + E(b) for all random variables a and b; and
4. E(za) = zE(a) if z is a constant.