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].

Because a point in space can be represented by a triple of real numbers, all geometric properties of spatial figures can be expressed in terms of real numbers. Hence one can theoretically understand geometry solely through analysis. But a true appreciation of geometry requires not only analytical technique but also intuition of geometric objects. The same holds for probability theory. The modern theory of probability is formulated in terms of measures and integrals and so is part of modern analysis from the logical viewpoint. But to really enjoy probability theory, one should grasp the orientation of development of the theory with intuitive insight into random phenomena. The purpose of this book is to explain basic probabilistic concepts rigorously as well as intuitively.

In Chapter 1 we restrict ourselves to trials with a finite number of outcomes. The concepts discussed here are those of elementary probability theory but are dealt with from the advanced standpoint. We hope that the reader appreciates how random phenomena are discussed mathematically without being annoyed with measure-theoretic complications.

In the subsequent chapters we expect the reader to be more or less familiar with basic facts in measure theory.

In Chapter 2 we discuss the properties of those probability measures that appear in this book.

In Chapter 3 we explain the fundamental concepts in probability theory such as events, random variables, independence, conditioning, and so on. We formulate these concepts on a perfect separable complete probability space. The additional conditions “perfectness” and “separability” are imposed to construct the theory in a more natural way. The reader will see that such conditions are satisfied in all problems appearing in applications.

In the standard textbook conditional probability is defined with respect to a-algebras of subsets of the sample space (Doob's definition). Here we first define it with respect to decompositions of the sample space (Kolmogorov's definition) to make it easier for the reader to understand its intuitive meaning and then explain Doob's definition and the relation between these two definitions.

Introduction

In this chapter we present some representation theorems for operators from Lp to Lp (μ), 0 < p < 1. The theorems have some important consequences; for example, we will show that a non-zero operator from Lp to Lp (μ), 0 < p < 1, is an isomorphism when restricted to Lp(A), for some set A of positive measure.

From Theorem 7.12 of the previous chapter, any non-zero endomorphism of Lp, 0 < p < 1, is an isomorphism on some infinite-dimensional subspace – and by Theorem 7.20 of the previous chapter, the subspace can be taken to be l p. We are now asserting considerably more. Our second assertion above trivially implies that a non-zero operator from Lp into Lp(μ) preserves a copy of since embeds isomorphically into Lp(A). Of course it also implies Pallaschke's original result that for 0 < p < 1, Lp admits no non-trivial compact endomorphisms.

Pallaschke's results on the endomorphisms of Lp, 0 < p < 1, appeared in 1973. A further step was taken by Berg- Peck-Porta [1973], who studied projections on L0. Kwapien [1973] characterized completely the operators from L0 to L0(μ).

Then Kalton [1978a] characterized completely the operators from Lp to Lp(μ)), 0 < p < 1, and derived a number of results on the structure of Lp, 0 < p < 1, including the above-mentioned one, as corollaries.