Note on the Tightness of the Metric on the Set of Complete Sub $\sigma$-Algebras of a Probability Space

Abstract

The purpose of this note is to show that the usual metric on the set of complete $\operatorname{sub} \sigma$-algebras of a probability space is very tight; a recent result from E. B. Boylan on equi-convergence of martingales follows and is thereby, we believe, better understood.