Rota's Fubini lectures: The first problem.

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 γ A 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 , F Ω , P) , with F Ω the sigma-algebra of Borel sets of a compact Hausdorff space Ω, and P a regular probability measure on F Ω , there is an MV-algebra A and a state σ of A such that (Ω , F Ω , P) ≅ (μ (A) , F μ (A) , γ A (σ)). [ABSTRACT FROM AUTHOR]