Some Applications of Boolean Valued Set Theory to Abstract Harmonic Analysis on Locally Compact Groups

Abstract

The main purpose of this paper is to extend Takeuti's [23] Boolean valued treatment of abstract harmonic analysis on locally compact abelian groups to locally compact groups (neither abelian nor compact in general). The distinctive feature of our approach, compared with traditional treatments of the subject, is that we can establish many important theorems without resort to direct integrals or to the theory of Banach algebras. By way of illustration, we will give such a proof of renowned Bochner's theorem. This paper is not intended to be exhaustive at all but hopefully to be suggestive. How far we can proceed in this direction yet remains to be seen.