Rota, probability, algebra and logic

Inspired by Rota's Fubini Lectures, we present the MV-algebraic extensions of various results in probability theory, first proved for boolean algebras by De Finetti, Kolmogorov, Carathéodory, Loomis, Sikorski and others. MV-algebras stand to Łukasiewicz infinite-valued logic as boolean algebras stand to boolean logic. Using Elliott's classification, the correspondence between countable boolean algebras and commutative AF C-algebras extends to a correspondence between countable MV-algebras and AF C-algebras whose Murray-von Neumann order of projections is a lattice. In this way, (faithful, invariant) MV-algebraic states are identified with (faithful, invariant) tracial states of their corresponding AF C-algebras. Faithful invariant states exist in all finitely presented MV-algebras. At the other extreme, working in the context of σ-complete MV-algebras we present a generalization of Carathéodory boolean algebraic probability theory.