Visible absorbing decompositions and uniqueness of invariant probabilities

Battle Count: 77

The paper is a foundational result in probability theory concerning Markov kernels. While it is categorized under q-fin.MF (Mathematical Finance) and the author references applications to ARCH and Markov switching models, the paper itself does not address any trading strategy, portfolio optimization, or market modeling directly. Its relevance is indirect: understanding uniqueness of invariant measures is a prerequisite for ergodic theorems used in long-run statistical properties of stochastic financial models. However, the result is too abstract and general to have immediate practical application in quantitative trading.

This is a pure mathematics paper with no algorithmic implementation. The 'methodology' is a mathematical proof using Jordan decomposition of signed measures. There is nothing to implement computationally. The theoretical result could inform the design of verification procedures for uniqueness of stationary distributions, but the paper does not propose any such procedure.

The paper is a purely theoretical mathematics article with complete proofs. All results are self-contained and verifiable from the definitions and lemmas provided. No computational experiments or data are involved. The proofs are elementary and use only standard measure-theoretic tools (Jordan decomposition, invariance properties).