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Measure and Integral: New foundations after one hundred years

  • Heinz KönigEmail author
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Abstract

The present article wants to describe the main ideas and developments in the theory of measure and integral in the course and at the end of the first century of its existence.

Keywords

Traditional abstract measures and Radon measures construction of measures after Carathéodory and Daniell-Stone inner regularity, construction via inner premeasures finite and infinite products the inner representation theorem projective limits and stochastic processes 

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References

  1. 1.
    N. Bourbaki, Intégration. Chap. 1–4, 2ième ed. Hermann 1965, Chap. 5, 2ième ed. Hermann 1967, Chap.IX, Hermann 1969, English translation Springer 2004.Google Scholar
  2. 2.
    C. Carathéodory, Über das lineare Mass von Punktmengen – eine Verallgemeinerung des Längenbegriffs. Nachr. K. Ges. Wiss. Göttingen, Math.-Nat.Kl. 1914, pp. 404–426. Reprinted in: Gesammelte Mathematische Schriften, Vol. IV, pp. 249–275. C.H.Beck 1956.Google Scholar
  3. 3.
    D.H. Fremlin, Measure Theory Vol. 1–4. Torres Fremlin 2000–2003 (in a numbered reference the first digit indicates its volume). "http://www.essex.ac.uk/maths/staff/fremlin/mt.htm".
  4. 4.
    P.R. Halmos, Measure Theory. van Nostrand 1950, Reprint Springer 1974.Google Scholar
  5. 5.
    J. Kisyński, On the generation of tight measures. Studia Math. 30(1968), 141–151.MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Kolmogorov(=ff), Grundbegriffe derWahrscheinlichkeitsrechnung. Springer 1933, Reprint 1973.Google Scholar
  7. 7.
    H. König, Measure and Integration: An Advanced Course in Basic Procedures and Applications. Springer 1997.Google Scholar
  8. 8.
    H. König, On the inner Daniell-Stone and Riesz representation theorems. Doc.Math. 5(2000), 301–315.MathSciNetzbMATHGoogle Scholar
  9. 9.
    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".
  10. 10.
    H. König, The (sup/super) additivity assertion of Choquet. Studia Math. 157(2003), 171–197.MathSciNetCrossRefGoogle Scholar
  11. 11.
    H. König, Stochastic processes on the basis of new measure theory. In: Proc. Conf. Positivity IV – Theory and Applications, TU Dresden 25–29 July 2005, pp. 79–92. Preprint No.107 under "http://www.math.uni-sb.de/PREPRINTS".
  12. 12.
    E. Marczewski, On compact measures. Fund.Math. 40(1953), 113–124.MathSciNetGoogle Scholar
  13. 13.
    D. Pollard and F. Topsøe, A unified approach to Riesz type representation theorems. Studia Math. 54(1975), 173–190.MathSciNetCrossRefGoogle Scholar
  14. 14.
    L. Schwartz, Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford Univ. Press 1973.zbMATHGoogle Scholar
  15. 15.
    F. Topsøe, Compactness in spaces of measures. Studia Math. 36(1970), 195–212.MathSciNetCrossRefGoogle Scholar
  16. 16.
    F. Topsøe, Topology and Measure. Lect.Notes Math. 133, Springer 1970.Google Scholar
  17. 17.
    F. Topsøe, Further results on integral representations. Studia Math. 55(1976), 239–245.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Fakultät für Mathematik und InformatikUniversität des SaarlandesSaarbrückenGermany

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