Radon-Nikodym theorems for nonnegative forms, measures and representable functionals

TL;DR

This paper establishes Radon-Nikodym theorems for nonnegative Hermitian forms and applies these results to prove related theorems for positive functionals and measures.

Contribution

It introduces Radon-Nikodym theorems for nonnegative Hermitian forms and extends these to positive functionals and measures.

Findings

Radon-Nikodym theorems for nonnegative Hermitian forms

Application to positive functionals and measures

Extension to both $\sigma$-additive and finitely additive measures

Abstract

The aim of this note is to establish two Radon--Nikodym type theorems for nonnegative Hermitian forms defined on a real or complex vector space. We apply these results to prove the known Radon--Nikodym theorems of the theory of representable positive functionals, $\sigma$-additive and finitely additive measures.