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→General Markov processes with countable state space: Add \forall i \in X in singleton to arbitrary |
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More generally take <math>X</math> and <math>Y</math> both countable and <math>\mathcal A = \mathcal P(X),\ \mathcal B = \mathcal P(Y)</math>.
Again a Markov kernel is defined by the probability it assigns to singleton sets for each <math>i \in X</math>
:<math>\kappa(B|i)=\sum_{j \in B}\kappa(\{j\}|i), \
We define a Markov process by defining a transition probability <math>P(j|i) = K_{ji}</math> where the numbers <math>K_{ji}</math> define a (countable) [[stochastic matrix]] <math>(K_{ji})</math> i.e.
:<math>\begin{align}
\sum_{j \in Y}K_{ji}&=1, \qquad &\forall i \in X.\\
\end{align}
</math>
We then define
:<math> \kappa(\{j\} | i) = K_{ji} = P(j|i), \qquad \forall i \in X, \quad \forall B \in \mathcal B</math>.
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