#### Recommended Citation

Published in *The Annals of Probability*, Volume 20, Issue 1, January 1, 1992, pages 421-454.

Copyright © 1992 Institute of Mathematical Statistics. The definitive version is available at http://dx.doi.org/10.1214/aop/1176989936.

*NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly*.

The definitive version is available at https://doi.org/10.1214/aop/1176989936.

#### Abstract

Starting with a Borel probability measure *P* on *X* (where *X* is a separable Banach space or a compact metrizable convex subset of a locally convex topological vector space), the class *F*(*P*), called the fusions of *P*, consists of all Borel probability measures on *X* which can be obtained from *P* by fusing parts of the mass of *P*, that is, by collapsing parts of the mass of *P* to their respective barycenters. The class *F*(*P*) is shown to be convex, and the ordering induced on the space of all Borel probability measures by *Q*≤ *P* if and only if *Q*ε *F*(*P*) is shown to be transitive and to imply the convex domination ordering. If *P* has a finite mean, then *F*(*P*) is uniformly integrable and *Q*≤ *P* is equivalent to *Q* convexly dominated by *P* and hence equivalent to the pair (*Q*, *P*) being martingalizable. These ideas are applied to obtain new martingale inequalities and a solution to a cost-reward problem concerning optimal fusions of a finite-dimensional distribution.

**URL:** http://digitalcommons.calpoly.edu/rgp_rsr/39