An Arbitrary Space of Elementary Events

  • Alexandr A. Borovkov
Part of the Universitext book series (UTX)


The chapter begins with the axiomatic construction of the probability space in the general case where the number of outcomes of an experiment is not necessarily countable. The concepts of algebra and sigma-algebra of sets are introduced and discussed in detail. Then the axioms of probability and, more generally, measure are presented and illustrated by several fundamental examples of measure spaces. The idea of extension of a measure is discussed, basing on the Carathéodory theorem (of which the proof is given in Appendix 1). Then the general elementary properties of probability are discussed in detail in Sect. 2.2. Conditional probability given an event is introduced along with the concept of independence in Sect. 2.3. The chapter concludes with Sect. 2.4 presenting the total probability formula and the Bayes formula, the former illustrated by an example leading to the introduction of the Poisson process.


  1. 28.
    Natanson, I.P.: Theory of Functions of a Real Variable. Ungar, New York (1961) Google Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Alexandr A. Borovkov
    • 1
  1. 1.Sobolev Institute of Mathematics and Novosibirsk State UniversityNovosibirskRussia

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