Some Finitely Additive Probability

Abstract

Lester E. Dubins and Leonard J. Savage have shown how to define a large family of finitely additive probability measures on the lattice of open sets of spaces of the form $X \times X \times \cdots$, where $X$, otherwise arbitrary, is assigned the discrete topology. This lattice does not include many of the sets which occur in the usual treatment of such probabilistic limit laws as the martingale convergence theorem, and in some unpublished notes Dubins and Savage conjectured that there might be a natural way to extend their measures to such sets. We confirm their conjecture here by showing that every set in the Borel sigma-field can be squeezed between an open and a closed set in the usual manner. It is then possible to generalize to this finitely additive setting many of the classical countably additive limit theorems. If assumptions of countable additivity are imposed, the extension studied here, when restricted to the usual product sigma-field, agrees with the conventional extension.