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By I. P. Natanson

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Basically, we will discuss only two large families of transition probabilities which will be useful throughout the book when dealing with examples of transition functions. The examples that follow are by no means the only important examples of transition probabilities. s. with probabilities, see, for example, the monograph by Lasota and Mackey [57], and the papers of Barnsley [5], Barnsley and Demko [6], Barnsley, Demko, Elton and Geronimo [7], Centore and Vrskay [17], Edalat [32], Lasota and Myjak [58– 61], and [62], Lasota and Yorke [64], Myjak and Szarek [80], Nicol, Sidorov and Broomhead [82], Stenflo [114–117], and [118], Szarek [120], Vrscay [125], and our papers [140,141], and [142].

1 of [143] in the case in which the transition probability is not necessarily Feller, but the elementary measures are standard elementary measures. We now return to the KBBY decomposition defined by the transition probability P . 5. X /. X /j 0; k k D 1g. Thus, €cp is the set of all x 2 €c such that "x 30 1 Transition Probabilities is a probability measure. Note that the total variation norm of an elementary (not necessarily standard) measure cannot exceed 1. 6. X /-measurable. TP/ x 2 €c . TP/ if x … €c is well-defined.

2; R/). 2; R/. 2; R/= €/L ! 2; R/= €/R ! 2; R/= €/R . 2; R/ because these elements are cosets. Thus, hO is the coset f h; hg defined by h. 2; R/= €/R are locally compact separable metrizable topological spaces. g; x€/ O 7! g; € x/ O 7! R/ are continuous. Accordingly, the functions uh and uh define Feller transition probabilities. L/ OO and uh , respectively. 2; R/= €/R /. R/ /, respectively. 2; R/= €/L / ! 2; R/= €/R / ! 2; R/= €/R . 2; R/= €/L / ! 2; R/= €/R / ! 8. 9. n; R/; n 2). n; R/). n; R/= €/L !

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