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An Application of Utrafilters to the Haar Measure

Volume 14, Number 1 (2012), 54 - 64

An Application of Utrafilters to the Haar Measure

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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 non-empty set $X$ is homeomorphic to the topological space of all nonzero multiplicative functionals in the first dual space $\ell_{\infty}^{\star}(X)$.