🤖 AI Summary
This study investigates the fundamental cause of uniqueness of invariant probability measures for Markov kernels on general state spaces. Employing a purely measure-theoretic approach, it introduces “indecomposability” as the core structural property ensuring uniqueness, reinterpreting classical irreducibility as a sufficient but not necessary condition. By leveraging the mutual singularity among distinct ergodic invariant measures and their common absolute continuity with respect to a reference measure, the work establishes a uniqueness criterion in standard Borel spaces that dispenses with assumptions of recurrence, regeneration, or kernel regularity. This framework both simplifies and extends classical ergodic theory, eliminating reliance on traditional dynamical systems tools.
📝 Abstract
We identify indecomposability as a key measure-theoretic underlying uniqueness of invariant probability measures for discrete-time Markov kernels on general state spaces. The argument relies on the mutual singularity of distinct invariant ergodic measures and on the observation that uniqueness follows whenever all invariant probability measures are forced to charge a common reference measure. Once existence of invariant probability measures is known, indecomposability alone is sufficient to rule out multiplicity. On standard Borel spaces, this viewpoint is consistent with the classical theory: irreducibility appears as a convenient sufficient condition ensuring indecomposability, rather than as a structural requirement for uniqueness. The resulting proofs are purely measure-theoretic and do not rely on recurrence, regeneration, return-time estimates, or regularity assumptions on the transition kernel.