In this final section we collect together, for ease of reference, many of those mathematical results which we have used in developing our results on Markov chains and their applications: these come from probability and measure theory, topology, stochastic processes, the theory of probabilities on topological spaces, and even number theory.

We have tried to give results at a relevant level of generality for each of the types of use: for example, since we assume that the leap from countable to general spaces or topological spaces is one that this book should encourage, we have reviewed (even if briefly) the simple aspects of this theory; conversely, we assume that only a relatively sophisticated audience will wish to see details of sample path results, and the martingale background provided requires some such sophistication.

Readers who are unfamiliar with any particular concepts and who wish to delve further into them should consult the standard references cited, although in general a deep understanding of many of these results is not vital to follow the development in this book itself.

Some measure theory

We assume throughout this book that the reader has some familiarity with the elements of measure and probability theory. The following sketch of key concepts will serve only as a reminder of terms, and perhaps as an introduction to some non-elementary concepts; anyone who is unfamiliar with this section must take much in the general state space part of the book on trust, or delve into serious texts such as Billingsley [37], Chung [72] or Doob [99] for enlightenment.