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The Sigma Algebra of Events

Chapter
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Part of the Springer Texts in Statistics book series (STS)

Abstract

In Chapter 1, we show that events are represented as subsets of the basic space Ω of all possible outcomes of the basic trial or experiment. Also, we see that it is highly desirable to consider logical operations on these subsets (complementation and the formation of unions and intersections). Very simple problems make desirable the treatment of sequences or other countable classes of events. Thus, if E is an event, the complementary set E c should be an event. If {E i : 1 ≤ i} is a sequence of events, then the union ∪ i=1 E i and the intersection ∩ i=1 E i should be events. On the other hand, there are technical mathematical reasons for not considering the class of events to be the class of all subsets of Ω. Basic spaces with an infinity of elements contain bizarre subsets that put a strain on the fundamental theory. We are thus led to consider some restrictions on the class of events, while still maintaining the desired flexibility and richness of membership. Long experience has shown that the proper choice is a sigma algebra. We wish to characterize these classes. Sigma algebras and many other important types of subclasses are characterized by various closure properties.

Keywords

Basic Space Closure Property Intersection Class Countable Union Finite Union 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  1. 1.Department of Mathematical SciencesRice UniversityHoustonUSA

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