Finitely additive functions in measure theory and applications

TL;DR

This paper extends the Radon-Nikodym derivative to finitely additive functions, exploring their applications in probability theory, stochastic calculus, and operator theory, providing new tools for analyzing non-sigma-additive measures.

Contribution

It introduces a precise extension of the Radon-Nikodym derivative for finitely additive functions and demonstrates its applications in probability, stochastic calculus, and operator analysis.

Findings

Extended Radon-Nikodym derivative for finitely additive functions.

Applications to $ ext{μ}$-Brownian motion and generalized Itô integrals.

Analysis of transition probability operators and composition operator adjoints.

Abstract

We consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory; in particular, to the study of $\mu$-Brownian motion, to stochastic calculus via generalized It\^o-integrals, and their adjoints (in the form of generalized stochastic derivatives), to systems of transition probability operators indexed by families of measures $\mu$, and to adjoints of composition operators.