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An abstract decomposition of measures and its many applications

Alessandro Milazzo, Pietro Siorpaes

TL;DR

This work revisits Dellacherie's decomposition theorem for positive measures and develops a broad, systematically applicable framework: it extends the decomposition to vector measures with control measures, applies it to the stochastic integral to obtain (and sharpen) semimartingale decompositions, and analyzes how the outputs depend on the inputs. It also establishes a spectral-measure variant, interprets the components as projections in both Riesz-space and metric-space senses, and finally generalizes the decomposition to strictly positive operators on Riesz spaces. Collectively, the results yield a unifying, versatile toolkit for decomposing measures, vector measures, stochastic processes, and operators, with clear connections to Lebesgue/Hahn–Jordan type decompositions and to projection-band structures. The findings provide new decompositions, elementary proofs for stochastic-analytic decompositions, and a principled operator-theoretic viewpoint that broadens Dellacherie’s classical result to a wide mathematical ecosystem.

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.

An abstract decomposition of measures and its many applications

TL;DR

This work revisits Dellacherie's decomposition theorem for positive measures and develops a broad, systematically applicable framework: it extends the decomposition to vector measures with control measures, applies it to the stochastic integral to obtain (and sharpen) semimartingale decompositions, and analyzes how the outputs depend on the inputs. It also establishes a spectral-measure variant, interprets the components as projections in both Riesz-space and metric-space senses, and finally generalizes the decomposition to strictly positive operators on Riesz spaces. Collectively, the results yield a unifying, versatile toolkit for decomposing measures, vector measures, stochastic processes, and operators, with clear connections to Lebesgue/Hahn–Jordan type decompositions and to projection-band structures. The findings provide new decompositions, elementary proofs for stochastic-analytic decompositions, and a principled operator-theoretic viewpoint that broadens Dellacherie’s classical result to a wide mathematical ecosystem.

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.
Paper Structure (8 sections, 19 theorems, 59 equations)

This paper contains 8 sections, 19 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Dellacherie's theorem for positive measures
  3. Examples of decompositions
  4. The decomposition theorem for vector measures
  5. The stochastic integral and a decomposition of semimartingales
  6. The spectral measure
  7. Dependence of $\mu_\mathcal{G}$ and $\mathcal{G}_\mu$ on $\mu$ and $\mathcal{G}$
  8. The decomposition theorem on Riesz spaces

Key Result

Theorem 1.1

Let $\mathcal{F}$ be a $\sigma$-algebra on $\Omega$, $\mu$ be a (positive and $\sigma$-additive) $\sigma$-finite measure on $(\Omega,\mathcal{F})$, and $\mathcal{G}\subseteq\mathcal{F}$ be non-empty and closed under countable unions. Then there exist a set $\bar{G}\in\mathcal{G}$ and two measures $\ where $\mu_\mathcal{G}(\bar{G}^c)=0$, and $\mu_\mathcal{G}^\perp(G)=0$ for every $G\in\mathcal{G}$.

Theorems & Definitions (56)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Corollary 3.1: Lebesgue decomposition
  • proof
  • Corollary 3.2: Decomposition into atomic and diffuse components
  • proof
  • ...and 46 more