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Our main goal in this paper is to prove that any transition probability P on a locally compact separable metric space (X, d) defines a Kryloff-Bogoliouboff-BeboutoffYosida (KBBY) ergodic decomposition of the state space (X, d). Our results extend and strengthen the results of Chap. 5 of Hernández-Lerma and Lasserre (Markov Chains and Invariant Probabilities, 2003) and extend our KBBY-decomposition for Markov-Feller operators that we have obtained in Chap. 2 of our monograph (Zaharopol in Invariant Probabilities of Markov-Feller Operators and Their Supports, 2005). In order to deal with the decomposition that we present in this paper, we had to overcome the fact that the LasotaYorke lemma (Theorem 1.2.4 in our book (op. cit.)) and two results of Lasota and Myjak (Proposition 1.1.7 and Corollary 1.1.8 of our work (op. cit.)) are no longer true in general in the non-Feller case.In the paper, we also obtain a "formula" for the supports of elementary measures of a fairly general type. The result is new even for Markov-Feller operators.We conclude the paper with an outline of the KBBY decomposition for a fairly large class of transition functions. The results for transition functions and transition probabilities seem to us surprisingly similar. However, as expected, the arguments needed to prove the results for transition functions are significantly more involved and are not presented here. We plan to discuss the KBBY decomposition for transition functions with full details in a small monograph that we are currently trying to write.

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Let (X, Σ,μ) be a measure space (where μ is a positive σ-additive measure) and let Lp(X,Σ,μ), 1≦p≦ + ∞ be the usual real Banach lattices.