Introduction

Abstract

Exposition of probability theory usually begins with a finite scheme. Following this tradition, consider a finite probabilistic space Ω:= 1,...,N. Three closely interrelated facts express in different ways the classical character of the probabilistic scheme:

• 1. the set of events A ⊂ Ω forms a Boolean algebra,

• 2. the set of probability distributions P = [p ₁,..., P _n] on ω is a simplex, that is a convex set in which each point is uniquely expressible as a mixture (convex linear combination) of extreme points,

• 3. the set of random variables X = [λ₁,..., λ_N] on Ω forms a commutative algebra (under pointwise multiplication).