The basic object of study in this book is the theory of discrete-time Markov processes or, briefly, Markov chains, defined on a general measurable space and having stationary transition probabilities.

The theory of Markov chains with values in a countable set (discrete Markov chains) can nowadays be regarded as part of classical probability theory. Its mathematical elegance, often involving the use of simple probabilistic arguments, and its practical applicability have made discrete Markov chains standard material in textbooks on probability theory and stochastic processes.

It is clear that the analysis of Markov chains on a general state space requires more elaborate techniques than in the discrete case. Despite these difficulties, by the beginning of the 1970s the general theory had developed to a mature state where all of the fundamental problems – such as cyclicity, the recurrence-transience classification, the existence of invariant measures, the convergence of the transition probabilities – had been answered in a satisfactory manner. At that time also several monographs on general Markov chains were published (e.g. Foguel, 1969 a; Orey, 1971; Rosenblatt, 1971; Revuz, 1975).

The primary motivation for writing this book has been in the recent developments in the theory of general (irreducible) Markov chains. In particular, owing to the discovery of embedded renewal processes, the ‘elementary’ techniques and. constructions based on the notion of regeneration, and common in the study of discrete chains, can now be applied in the general case.

Since P.J. Cohen's proof of the independence of the continuum hypothesis from the other axioms of set theory, there has been a remarkable flowering of similar results, ranging over all those branches of mathematics which deal in propositions of a similar level. Many of these are specific constructions dealing with individual problems. But some of the alternative models of set theory that have been developed provide answers to several questions. In terms of the number and variety of their uses, two are at present outstanding: model Δ of Gödel 40, and the models of Solovay & Tennenbaum 71. Each of these was constructed with the aim of showing the consistency of a particular hypothesis (in the former, the continuum hypothesis; in the latter, Souslin's hypothesis); but in each case an enormous number of unexpected further properties has emerged.

The structure of model Δ is such that, although it can be regarded as ordinary mathematics with one extra axiom added (the axiom of constructibility, or ‘V = L’), it is not possible to make deductions from this axiom without appealing to non-trivial ideas from mathematical logic; so that the non-logician who wishes to examine its consequences must work from one level lower (e.g. from R.B. Jensen's principle ♦). But the most useful properties of the Solovay–Tennenbaum model (or, rather, models) are relatively accessible, being derived by conventional arguments from Martin's axiom, ‘MA’, ‘m = c’, in one of its variations.