An abstract decomposition of measures and its many applications

TL;DR

This paper explores a fundamental measure decomposition, extends it to vector measures, improves existing semimartingale decompositions, and generalizes to positive operators, revealing deep structural insights.

Contribution

It introduces new measure decompositions, extends Dellacherie's result to vector measures, and generalizes to positive operators, advancing measure theory and stochastic process analysis.

Findings

New measure decompositions derived from Dellacherie's result

Extended decomposition to controlled vector measures

Improved Bichteler's semimartingale decomposition

Abstract

We consider a little-known abstract decomposition result for positive measures due to Dellacherie, and show that it yields many decompositions of measures, several of which are new. We then extend Dellacherie's result to (controlled) vector measures, and apply it to obtain a decomposition of semimartingales due to Bichteler, on which we improve. Then, we investigate how the outputs of the decomposition depend on its inputs, in particular characterising the two elements of the decomposition as projections in the sense of Riesz spaces and of metric spaces. Finally, we prove a decomposition theorem for strictly positive operators on Riesz spaces which generalises Dellacherie's Theorem.