On orderable set functions and continuity I

A set functionv (which is not necessarily additive) on a measurable spaceI is called orderable if for each measurable order ℛ onI there is a measureϱ ^ℛ ν(J) =ν(J) onI such that for all subsetsJ ofI that are initial segements,ϱ ^ℛ ν. Properties such as nonatomicity, nullness of sets, and weak continuity are shown to be inherited from orderable set functionsv toϱ ^ℛ ν and vice versa. A characterization of set functions which are absolutely continuous (with respect to some positive measure) in the set of orderable set functions is also given. Reporduction of this report was partially supported by the Office of Naval Research under contract N-000 14-67-0112-0011; The U.S. Atomic Energy Commission contract AT (04-3)-326-PA #18; and The National Science Foundation, Grant GP 31393X. Reproduction in whole or in part is permitted for any purposes of the United States Government. This document has been approved for public release and sale; its distribution is unlimited. Research of this report was carried out at Tel Aviv University.