It is generally known
that if X is a σ-compact metric space, then every Borel measure on X is
regular. It is not difficult to prove a slightly stronger result, namely that
the same conclusion holds if X is a Hausdorff space in which every open
subset is σ-compact (I.6 below). The converse is not generally true, even for
compact Hausdorff spaces; a counter-example appears here under IV. 1.
However, it will be shown in §II that every nondegenerate Borel measure on a
nondiscrete locally compact group is regular if and only if the group is σ-compact
and metrizable. A similar theorem, proved in §III, holds for dyadic spaces:
every Borel measure on such a space is regular if and only if the space is
metric.
The result for groups depends on two structure theorems which are proved here:
every nonmetrizable compact connected group contains a nonmetrizable
connected Abelian subgroup (II.10), and every nonmetrizable locally compact
group contains a nonmetrizable compact totally disconnected subgroup
(II.11).
