An Application of Ultrafilters to the Haar Measure

Abstract

In this article we will use ultrafilter theory to present a modified proof that a locally compact group with a countable basis has a left invariant and a right invariant Haar measure. To facilitate this result, we shall first show that the topological space consisting of all ultrafilters on a nonempty set $X$ is homeomorphic to the topological space of all nonzero multiplicative functionals in the first dual space $\ell_{\infty}^{\star}(X)$.