Stochastic processes in terms of inner premeasures

  • Heinz KönigEmail author


In a recent paper the author used his work in measure and integration to obtain the projective limit theorem of Kolmogorov in a comprehensive version in terms of inner premeasures. In the present paper the issue is the influence of the new theorem on the notion of stochastic processes. It leads to essential improvements in the foundation of special processes, of the Wiener process in the previous paper and of the Poisson process in the present one. But it also forms the basis for a natural redefinition of the entire notion. The stochastic processes in the reformed sense are in one-to-one correspondence with the traditional ones in case that the state space is a Polish topological space with its Borel σ algebra, but the sizes and procedures are quite different. The present approach makes an old idea of Kakutani come true, but with due adaptations.


Stochastic processes Wiener process Poisson process inner premeasures maximal inner extensions projective families Polish spaces 


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  1. [1]
    H. Bauer:Wahrscheinlichkeitstheorie. 4th ed. de Gruyter 1991, English translation 1996.Google Scholar
  2. [2]
    N. Bourbaki: Intégration, Chap. IX. Hermann 1969.zbMATHGoogle Scholar
  3. [3]
    C. Dellacherie, P. A. Meyer: Probability and Potential, North-Holland 1978.Google Scholar
  4. [4]
    J. L. Doob: Stochastic processes depending on a continuous parameter, Trans. Amer. Math. Soc., 42, (1937), 107–140.MathSciNetCrossRefGoogle Scholar
  5. [5]
    J. L. Doob: Regularity properties of certain families of chance variables, Trans.Amer. Math.Soc., 47, (1940), 455–486.MathSciNetCrossRefGoogle Scholar
  6. [6]
    J. L. Doob: Probability in function spaces, Bull. Amer. Math. Soc., 53, (1947), 15–30.MathSciNetCrossRefGoogle Scholar
  7. [7]
    J. L. Doob: Stochastic Processes, Wiley 1953.Google Scholar
  8. [8]
    D. H. Fremlin: Measure Theory, Vol.1–4, Torres Fremlin 2000–2003 (in the references the first digit of an item indicates its volume). fremlin/mt.htm.Google Scholar
  9. [9]
    W. Hackenbroch, A. Thalmaier: Stochastische Analysis, Teubner 1994.Google Scholar
  10. [10]
    A. Kolmogorov (=Kolmogoroff): Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer 1933, Reprint 1973.Google Scholar
  11. [11]
    H. König: Measure and Integration: An advanced Course in basic Procedures and Applications, Springer 1997.Google Scholar
  12. [12]
    H. König: The product theory for inner premeasures, Note di Matematica, 17, (1997), 235–249.Google Scholar
  13. [13]
    H. König: Measure and integration: An attempt at unified systematization, Rend.Istit. Mat. Univ. Trieste, 34, (2002), 155–214. Preprint No.42 under http://www.math.uni-sb. de/PREPRINTS.Google Scholar
  14. [14]
    H. König: Projective limits via inner premeasures and the true Wiener measure, Mediterranean J.Math., 1, (2004), 3–42.Google Scholar
  15. [15]
    K. R. Stromberg: Probability for Analysts, Chapman & Hall 1994.Google Scholar

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© Springer Basel 2012

Authors and Affiliations

  1. 1.Fakultät für Mathematik und InformatikUniversität des SaarlandesSaarbrückenGermany

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