Vol. 28, No. 1, 1969

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Regular and irregular measures on groups and dyadic spaces

Harold L. Peterson, Jr.

Vol. 28 (1969), No. 1, 173–182

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).

Mathematical Subject Classification
Primary: 22.20
Secondary: 28.00
Received: 26 March 1968
Published: 1 January 1969
Harold L. Peterson, Jr.