Another ‘short’ proof of the Riesz representation theorem

Abstract

1. Introduction. In [2], a short synthetic proof of the Riesz representation theorem was given; this used the Hahn-Banach theorem, the Stone-Čech compactification of a discrete space and the Caratheodory extension procedure for measures. In this note, we show how the theorem can be proved using ultrapowers in place of the Stone-Čech compactification. We also describe how the proof can be expressed in a non-standard way (a rather different non-standard proof has been given by Loeb [4]).