Probability-1 pp 159-371 | Cite as

Mathematical Foundations of Probability Theory

  • Albert N. Shiryaev
Part of the Graduate Texts in Mathematics book series (GTM, volume 95)


The theory of probability, as a mathematical discipline, can and should be developed from axioms in exactly the same way as Geometry and Algebra. This means that after we have defined the elements to be studied and their basic relations, and have stated the axioms by which these relations are to be governed, all further exposition must be based exclusively on these axioms, independent of the usual concrete meaning of these elements and their relations.


  1. [1]
    M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Courier Dover Publications, New York, 1972.Google Scholar
  2. [9]
    P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968.Google Scholar
  3. [10]
    P. Billingsley. Probability and Measure. 3rd ed. New York, Wiley, 1995.Google Scholar
  4. [28]
    N. Dunford and J. T. Schwartz, Linear Operators, Part 1, General Theory. Wiley, New York, 1988.Google Scholar
  5. [31]
    W. Feller. An Introduction to Probability Theory and Its Applications, vol. 2, 2nd ed. Wiley, New York, 1966.Google Scholar
  6. [35]
    I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. Academic Press, New York, 1994.Google Scholar
  7. [39]
    P. R. Halmos. Measure Theory. Van Nostrand, New York, 1950.Google Scholar
  8. [43]
    J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin–Heidelberg, 1987.Google Scholar
  9. [51]
    A. N. Kolmogorov. Foundations of the Theory of Probability. Chelsea, New York, 1956; second edition [Osnovnye poniatiya Teorii Veroyatnosteĭ]. “Nauka”, Moscow, 1974.Google Scholar
  10. [52]
    A. N. Kolmogorov and S. V. Fomin. Elements of the Theory of Functions and Functionals Analysis. Graylok, Rochester, 1957 (vol. 1), 1961 (vol. 2); sixth edition [Elementy teorii funktsiĭ i funktsionalnogo analiza]. “Nauka” Moscow, 1989.Google Scholar
  11. [64]
    M. Loève. Probability Theory. Springer-Verlag, New York, 1977–78.Google Scholar
  12. [65]
    E. Lukacs. Characteristic functions. 2nd edition. Briffin, London, 1970.Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Albert N. Shiryaev
    • 1
  1. 1.Department of Probability Theory and Mathematical StatisticsSteklov Mathematical Institute and Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations