Construction of regular non-atomic strictly-positive measures in second-countable non-atomic locally compact Hausdorff spaces

Jason Bentley


This paper presents a constructive proof of the existence of a regular non-atomic strictly-positive measure on any second-countable non-atomic locally compact Hausdorff space. This construction involves a sequence of finitely-additive set functions defined recursively on an ascending sequence of rings of subsets with a set function limit that is extendable to a measure with the desired properties. Non-atomicity of the space provides a meticulous way to ensure that the set function limit is σ-additive.


regular measure; non-atomic measure; strictly-positive measure; locally compact spaces; non-atomic spaces; Polish spaces; regular spaces; second-countable spaces

1. D.L. Cohn, Measure Theory, 2nd Edition, Birkhäuser, New York, 2013.
2. P. Mikusiński, On the completion of measures, Arch. Math. (Basel) 50 (1988), 259–263.
3. M.E. Munroe, Introduction to Measure and Integration, Addison-Wesley, Cambridge, Mass., 1953.
4. W. Rudin, Real and Complex Analysis, 3rd Edition, McGraw-Hill, New York, 1987.

Published : 2022-03-22

BentleyJ. (2022). Construction of regular non-atomic strictly-positive measures in second-countable non-atomic locally compact Hausdorff spaces. Annales Mathematicae Silesianae, 36(1), 15-25. Retrieved from

Jason Bentley
Department of Mathematics, University of Central Florida, USA  United States

The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.

  1. License
    This journal provides immediate open access to its content under the Creative Commons BY 4.0 license ( Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license.
  2. Author’s Warranties
    The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s.
  3. User Rights
    Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor.
  4. Co-Authorship
    If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.