Endogenous Measures and Refinement Dynamics on Finite {\sigma}-Algebra Systems

TL;DR

This paper investigates endogenous probability measures on finite sigma-algebra systems, exploring their existence, structural properties, and the dynamical effects of refinement operators.

Contribution

It introduces the concept of endogenous measures invariant under refinement and analyzes their structural and dynamical properties on finite systems.

Findings

Existence of endogenous measures on finite sigma-algebra systems

Structural properties of these measures

Refinement operators induce a natural dynamical structure

Abstract

We study systems of {\sigma}-algebras ordered by refinement and introduce the notion of an endogenous probability measure, invariant under admissible refinement transformations. We prove existence and structural properties of such measures on finite systems and show how refinement operators induce a natural dynamical structure on the lattice of {\sigma}-algebras.