Discrete and Compact Groups

Abstract

This chapter is devoted to harmonic analysis on some compact abelian groups other than the circle group. We shall construct Haar measure for compact abelian groups, then prove the Pontryagin duality theorem for compact and discrete abelian groups, a theorem of Minkowski, Kolmogorov’s extension theorem, and finally the Banach-Steinhaus theorem as a consequence of a theorem of Steinhaus about the set of distances between points in a set of positive measure.