International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 2, Pages 277-282
doi:10.1155/S0161171293000328
A metric space associated with probability space
Department of Mathematics, University of Saskatchewan, Sask., Saskatoon S7N 0W0, Canada
Received 7 January 1992
Copyright © 1993 Keith F. Taylor and Xikui Wang. 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.
Abstract
For a complete probability space (Ω,∑,P), the set of all complete sub-σ-algebras
of ∑, S(∑), is given a natural metric and studied. The questions of when S(∑) is compact or
connected are awswered and the important subset consisting of all continuous sub-σ-algebras is
shown to be closed. Connections with Christensen's metric on the von Neumann subalgebras of a
Type II1-factor are briefly discussed.