Vol. 28, No. 1, 1969

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Online Archive
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author index
To appear
Other MSP journals
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.