Introduction

In this chapter we discuss the density and tangency structure of s-sets in ℝn when s is an integer. We know from Corollary 2.10 that an s-set splits into a regular part and an irregular part, and we find that these two types of set exhibit markedly different properties. One of our aims is to characterize regular sets as subsets of countable unions of rectifiable curves or surfaces, and thus to relate the measure theoretic and the descriptive topological ideas.

We present in detail the theory of linearly measurable sets or 1-sets in ℝ2. This work is almost entirely due to Besicovitch (1928a, 1938), the latter paper including some improved proofs as well as further results. Most of his proofs seem hard to better except in relatively minor ways and, hopefully, in presentation. Certainly, some of the geometrical methods used by Besicovitch involve such a degree of ingenuity that it is surprising that they were ever thought of at all. Some of the work in this chapter is also described in de Guzman (1981).

Curves and continua

Regular 1-sets and rectifiable curves are intimately related. Indeed, a regular 1-set is, to within a set of measure zero, a subset of a countable collection of rectifiable curves. This section is devoted to a study of curves, mainly from a topological viewpoint and in relation to continua of finite linear measure. Here we work in ℝn as the theory is no more complicated than for plane curves.

A curve (or Jordan curve) Γ is the image of a continuous injection ψ:[a,b]→ℝn, where [a,b]⊂:ℝ is a closed interval. Any curve is a continuum, that is, a compact connected set.

Introduction

The Kakeya problem has an interesting history. In 1917 Besicovitch was working on problems on Riemann integration, and was confronted with the following question: if f is a Riemann integrable function defined on the plane, is it always possible to find a pair of orthogonal coordinate axes with respect to which ∫ f(x,y)dx exists as a Riemann integral for all y, and with the resulting function of y also Riemann integrable? Besicovitch noticed that if he could construct a compact set F of plane Lebesgue measure zero containing a line segment in every direction, this would lead to a counter-example: For assume, by translating F if necessary, that F contains no segment parallel to and rational distance from either of a fixed pair of axes. Let f be the characteristic function of the set F0 consisting of those points of F with at least one rational coordinate. As F contains a segment in every direction on which both F0 and its complement are dense, there is a segment in each direction on which f is not Riemann integrable. On the other hand, the set of points of discontinuity of F is of plane measure zero, so f is Riemann integrable over the plane by the wellknown criterion of Lebesgue.

Besicovitch (1919) succeeded in constructing a set, known as a ‘Besicovitch set’, with the required properties. Owing to the unstable situation in Russia at the time, his paper received limited circulation, and the construction was later republished in Mathematische Zeitschrift (1928).

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.

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.