Classical probability theory

In a classical probabilistic space you have events and a probability measure:

(Ω, Σ, P) .

But you want numerical data, so you study random variables: functions

f : Ω \mapsto C

such that

f^{- 1} (A) \in Σ

for every open set $A = {z \in C : a < R e [z] < b}$ and

s u p_{Ω} {f (ω)} < \infty .

They constitute the algebra $L^{\infty} (Ω, Σ, P)$ , and the expected value for any random variable $g$ , defined by

E_{P} [g] = \int_{Ω} g (ω) \cdot P (ω)

plays the role of a linear functional

E_{P} : L^{\infty} (Ω, Σ, P) \mapsto C

In fact, it is not only an algebra, but a commutative von Neumann algebra, and $E_{P}$ is what it is called a faithful, normal state (see Quantum Probability Theory, from Hans Maasen, for details).
There is a theorem called the Gelfand-Naimark theorem which applied to a von Neumann algebra with a normal, faithful state let us recover the original probabilistic space. That is to say: all the data is inside the von Neumann algebra and the linear functional.

Sketch of the construction: if we begin with an algebra $A$ (it could be $L^{\infty} (Ω, Σ, P)$ or not) we recover a $σ$ -algebra $Σ$ by selecting all the $p \in A$ such that $p^{2} = p = p^{*}$ . Think if $p \in L^{\infty}$ is like that, $p (ω) = 0$ or $p (ω) = 1$ , and $p$ works like and indicator function of a set $A_{p} \in Σ$ . The inclusion $\subseteq$ relation in $Σ$ is recovered by means of the relation: $p \subseteq q$ iff $p \cdot q = p$ . And so we can recover the elementary events and the sample space $Ω$ .