Boolean Representations in Functional Analysis

Abstract

We have already convinced ourselves that the Boolean-valued universe V^(B) associated with a fixed Boolean algebra B is one of the arenas where mathematical events occur. Indeed, by virtue of the transfer and maximum principles, in V^(B) there are numbers and groups, Lebesque and Riemann integrals, the Radon-Nikodym theorems are fulfilled, and the Jordan expansion of a matrix is implementable. The elementary technique of descents and ascents, which we got acquainted with when considering algebraic systems, shows each of mathematical objects in V^(B) Mo be a realization of an analogous classical object with an additional structure determined by the algebra B. In particular, this consideration refers to functional-analytical objects as well.