International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 4, Pages 801-808

A Lebesgue decomposition for elements in a topological group

Thomas P. Dence

Department of Mathematics, Bowling Green State University, Firelands Campus, Huron 44839, Ohio, USA

Received 3 May 1979; Revised 29 February 1980

Copyright © 1980 Thomas P. Dence. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Our aim is to establish the Lebesgue decomposition for strongly-bounded elements in a topological group. In 1963 Richard Darst established a result giving the Lebesgue decomposition of strongly-bounded elements in a normed Abelian group with respect to an algebra of projection operators. Consequently, one can establish the decomposition of strongly-bounded additive functions defined on an algebra of sets. Analagous results follow for lattices of sets. Generalizing some of the techniques yield decompositions for elements in a topological group.