Uniqueness of invariant measures as a structural property of markov kernels

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.