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.

I come at last to results which need the full strength of the cardinal m; that is to say, which involve partially ordered sets which are ccc but may not satisfy Knaster's condition. I divide these into two sections on combinatorics (§§41–42) and two on general topology (§§43–44).

Combinatorics I

I base this section on the result that if m > ω1 then every ccc partially ordered set satisfies Knaster's condition [41Ab]. This is already enough to prove that the product of ccc spaces is ccc [41E], so that Souslin's hypothesis is true [41D, 41F–G], and enables us to apply the results of §31 to ccc sets [41B–C]. I give a result in the partition calculus [41H] with some important corollaries. I conclude with a version of ‘Devlin's axiom’ [41K] and a description of some principles apparently weaker than m > ω1 [41L].

Theorem

[m > ω1] Let P be an upwards-ccc partially ordered set.

1. (a) If is a family in P, there is an uncountable A ξ ω1 such that {pξ:ξ∈A} is upwards-centered in P.

2. (b) P satisfies Knaster's condition upwards.

Proof (a) For ξ, < ω1 set

Qξ = {p:p∈P, ∃η≥ξ such that p ≥ pη}.

Then each Qξ is up-open in P, and Qξ ξ Qη whenever η ≤ ξ. Now there is a ζ < ω1 such that Qξ is cofinal with Qξ for every ξ ≥ ζ. P?

In this appendix I write out the definitions and theorems which I use in the pages above, and for which no natural place presented itself in the main line of the exposition. In general I give proofs only when I have been unable to find satisfactory references in hard covers. I hope that the index will prove adequate and that there will be no need for you to read systematically through this appendix; but perhaps a preliminary glance at §A1 will be useful. Some of the material which you might look for here is in §12.

Notation

Here I list some of the special symbols I use, and indicate the ways in which I think of some of the fundamental concepts of set theory. I have tried to express these in terms which are readily translatable into the formulae of any conventional description of Zermelo–Fraenkel set theory, though it will be clear that this particular framework is not the only possible one. Note that I use the axiom of choice without scruple and without comment.

Reserved symbols

1. (a) N, Z, Q, R represent respectively the sets of non-negative integers, integers, rational numbers and real numbers.

2. (b) ω is the first infinite ordinal. ω1 is the first uncountable ordinal. c = 2ω = #(R), the cardinal of the continuum, κ and λ always stand for cardinals, m, mK and p stand for the special cardinals defined in §11.

3. […]

This chapter is devoted to results which involve the cardinal p. The definition of p in terms of families of subsets of N makes it plain that we must expect to be limited to contexts in which countable sets play a dominant role. Thus in Theorem 21A, the underlying set X must be countable; in §§22–23, we deal mainly with second-countable spaces; in §24, we have separable spaces; and in §25 we work with spaces which have associated second-countable topologies. It is, however, worth noting that several of the arguments can reach surprisingly far. Thus 21G can apply to arbitrary subsets of c2 and 24K does not mention any restriction on cardinality, though of course such a restriction is present.

Combinatorics

The axiom p = c is, of course, a tool for finding (or, rather, an excuse for declaring the existence of) subsets of N; and it is natural to give priority to its set-theoretic consequences, even though the distinction between these and its topological consequences is not always sharp. I begin with a portmanteau theorem [21A] which embodies most of the straight forward ways of using the principle P(k) itself. I use this to prove first that 2K ≤ c for K < p [21C] and then to give conditions sufficient to make every subset of X × Y attainable in two steps from sequences of ‘rectangles’ [21G]. I conclude with remarks on four further topics: a lemma in the partition calculus [211], R.B. Jensen's principle ♦ [21J], the possible values of p [21K] and Hausdorff's gap [21L].