Irregular invariant measures related to Haar measure

Abstract:

Let G be a locally compact nondiscrete group, and let ${\nu _1}$ be a Haar measure on an open subgroup of $G$. It is not hard to show that ${\nu _1}$ must be the restriction of a Haar measure $\nu$ on all of $G$. Here we show that there exists a translation invariant measure $\mu$ (found by extending ${\nu _1}$ to the cosets of $H$ in a natural way) which agrees with $\nu$ on, for example, $(\nu )$ $\sigma$-finite sets, open sets, and subsets of $H$. Although $\nu$ can be computed from $\mu$ in a relatively simple manner, the two measures are not equal in general. In fact, there is an extreme case, namely when $H$ is not $\sigma$-compact and has uncountably many cosets, in which $\mu$ fails very badly to be regular—there are closed sets on which $\mu$ is not inner regular and (other) closed sets on which $\mu$ is not outer regular. One condition sufficient for this extreme case to be possible is when $G$ is Abelian and not $\sigma$-compact.