International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 4, Pages 801-808
doi:10.1155/S0161171280000609
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
Abstract
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.