Achim Klenke - Probability Theory

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Achim Klenke Universitext Probability Theory 2nd ed. 2014 A Comprehensive Course 10.1007/978-1-4471-5361-0_1 Springer-Verlag London 2014

1. Basic Measure Theory

Achim Klenke 1

(1)

Institut fr Mathematik, Johannes Gutenberg-Universitt Mainz, Mainz, Germany

Abstract

In this chapter, we lay the measure theoretic foundations of probability theory. We introduce the classes of sets (semirings, rings, algebras, -algebras) that allow for a systematic treatment of events and random observations. Using the measure extension theorem, we construct measures, in particular probability measures on -algebras. Finally, we define random variables as measurable maps and study the -algebras generated by certain maps.

In this chapter, we introduce the classes of sets that allow for a systematic treatment of events and random observations in the framework of probability theory. Furthermore, we construct measures, in particular probability measures, on such classes of sets. Finally, we define random variables as measurable maps.

1.1 Classes of Sets

In the following, let be a nonempty set and let (set of all subsets of ) be a class of subsets of . Later, will be interpreted as the space of elementary events and will be the system of observable events. In this section, we introduce names for classes of subsets of that are stable under certain set operations and we establish simple relations between such classes.

Definition 1.1

A class of sets is called

• -closed (closed under intersections) or a - system if whenever ,
• --closed (closed under countable intersections) if for any choice of countably many sets ,
• -closed (closed under unions) if whenever ,
• --closed (closed under countable unions) if for any choice of countably many sets ,
• -closed (closed under differences) if whenever , and
• closed under complements if for any set .

Definition 1.2

(-algebra)

A class of sets is called a - algebra if it fulfills the following three conditions:

(i)

.

(ii)

is closed under complements.

(iii)

is closed under countable unions.

Sometimes a -algebra is also named a -field. As we will see, we can define probabilities on -algebras in a consistent way. Hence these are the natural classes of sets to be considered as events in probability theory.

Theorem 1.3

If is closed under complements , then we have the equivalences

Proof

The two statements are immediate consequences of de Morgans rule (reminder: ). For example, let be --closed and let . Hence

Thus is --closed. The other cases can be proved similarly.

Theorem 1.4

Assume that is - closed . Then the following statements hold :

(i)

is - closed .

(ii)

If in addition is -- closed , then is -- closed .

(iii)

Any countable ( respectively finite ) union of sets in can be expressed as a countable ( respectively finite ) disjoint union of sets in .

Proof

(i) Assume that . Hence also .

(ii) Assume that . Hence

(iii) Assume that . Hence a representation of

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