Chapter 1 Generalities This chapter is of an introductory nature. We start with recalling some basic probabilistic notions and facts in Sec. 1. Actually, the reader is supposed to be familiar with the material of this rather short section, which in no way is intended to be a systematic introduction to probability theory. All missing details can be found, for instance, in excellent books by R. Dudley [Du] and D. Stroock [St]. In Sec. 2 we discuss measures on Polish spaces. Quite often this subject is also included in courses on probability theory. Sec. 3 is devoted to the notion of random process, and in Sec. 4 we discuss the relation between continuous random processes and measures on the space of continuous functions. 1. Some selected topics from probability theory The purpose of this section is to remember some familiar tunes and get warmed up. We just want to refresh our memory, recall some standard notions and facts, and introduce the notation to be used in the future. Let Vt be a set and T a collection of its subsets. 1. Definition. We say that T is a a-field if (i) ft e T, (ii) for every A\,..., An,... such that An G T", we have (J n An G T', (hi) if A G T, then A c : = f i \ i e f . In the case when T is a a-field the couple (^,JT) is called a measurable space, and elements of T are called events. 2. Example. Let Q be a set. Then T := {0, fi} is a a-field which is called the trivial cr-field. 1
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