CHAPTER II - SOME CLASSICAL THEORY

Summary

This chapter is a reminder of what every probabilist should know, with the emphasis on things that tend to be neglected. The considerable length of the chapter—and it is now much more extensive than in the first edition—should be sufficient guarantee that ‘reminder’ is used in the usual ‘courtesy’ sense! Because things are now developed in strictly logical order, you may sometimes have to wait a little time for applications. (We very occasionally cheat just a little in Exercises by using things that you may feel are not yet proved with full rigour, but we always clear these up later.) Exercises are very much part of the text—please do them!

So many reminders of standard definitions are included that we break with our usual definition format except when we wish to give special emphasis to particularly important material that may not be so familiar.

BASIC MEASURE THEORY

The basic results of measure theory are summarised here, with commentary, but mostly without proofs. A full account, with all results proved, may be found, for example, in Williams [15], referred to as [W] throughout this chapter; that account has the advantage that its notation and terminology are the same as those used here. Neveu [1] is a marvellous account of measure theory for probabilists; and, for the definitive account of the full theory, including Choquet capacitability theory (which is needed for the Debut and Section Theorems), see Volume 1 of Dellacherie and Meyer [1].