The existence of regular conditional probabilities for Markov kernels

In this work, considering a Markov kernel (also called a stochastic kernel or transition probability) as a generalization of the concepts of the sigma-field and the random variable, the concept of a conditional distribution (or regular conditional probability) of a Markov kernel given another is introduced. The existence of such a conditional distribution is proved under standard regularity conditions concerning the measurable spaces involved. Besides this, a characterization of the independence of Markov kernels is obtained in terms of conditional distributions. The work ends with a description of such a conditional distribution when densities are available. (C) 2012 Elsevier B.V. All rights reserved.