Rota's Fubini lectures: The first problem

Abstract

In his 1998 Fubini Lectures, Rota discusses twelve problems in probability that “no one likes to bring up”. The first problem calls for a revision of the notion of a sample space, guided by the belief that mention of sample points in a probabilistic argument is bad form and that a “pointless” foundation of probability should be provided by algebras of random variables.In 1958 Chang introduced MV-algebras to prove the completeness theorem of Łukasiewicz logic Ł∞. The aim of this paper is to show that MV-algebras provide a solution of Rota's first problem.The adjunction between MV-algebras and unital commutative C*-algebras equips every MV-algebra A with a natural ring structure, as advocated by Nelson for algebras of random variables. The closed compact set S(A)⊆[0,1]A of finitely additive probability measures on A (the states of A) coincides with the set of [0,1]-valued functions on A whose finite restrictions are consistent in de Finetti's sense. MV-algebras and Ł∞ thus provide the framework for a generalization (known as ŁIPSAT) of Boole's probabilistic inference problem, and its modern reformulation known as probabilistic satisfiability, PSAT. We construct an affine homeomorphism γA of S(A) onto the weakly compact space of regular Borel probability measures on the maximal spectral space μ(A). The latter is the most general compact Hausdorff space. As a consequence, for every Kolmogorov probability space (Ω,FΩ,P), with FΩ the sigma-algebra of Borel sets of a compact Hausdorff space Ω, and P a regular probability measure on FΩ, there is an MV-algebra A and a state σ of A such that (Ω,FΩ,P)≅(μ(A),Fμ(A),γA(σ)).