Skip to main content

Quasi-Invariant and Pseudo-Differentiable Measures with Values in Non-Archimedean Fields on a Non-Archimedean Banach Space

Abstract

Quasi-invariant and pseudo-differentiable measures on a Banach space X over a non-Archimedean locally compact infinite field with a non-trivial valuation are defined and constructed. Measures are considered with values in non-Archimedean fields, for example, the field Q p of p-adic numbers. Theorems and criteria are formulated and proved about quasi-invariance and pseudo-differentiability of measures relative to linear and non-linear operators on X. Characteristic functionals of measures are studied. Moreover, the non-Archimedean analogs of the Bochner-Kolmogorov and Minlos-Sazonov theorems are investigated. Infinite products of measures are considered and the analog of the Kakutani theorem is proved. Convergence of quasi-invariant and pseudo-differentiable measures in the corresponding spaces of measures is investigated.

This is a preview of subscription content, access via your institution.

REFERENCES

  1. 1.

    I. Ya. Aref’eva, B. Dragovich, and I. V. Volovich, “On the p-adic summability of the anharmonic oscillator,” Phys. Lett., B 200, 512–514 (1988).

    Google Scholar 

  2. 2.

    A. H. Bikulov and I. V. Volovich, “p-Adic Brownian motion,” Izv. Ros. Akad. Nauk, Ser. Mat., 61, No.3, 75–90 (1997).

    Google Scholar 

  3. 3.

    S. Bosch, U. Guntzer, and R. Remmert, Non-Archimedean Analysis, Springer, Berlin (1984).

    Google Scholar 

  4. 4.

    V. I. Bogachev and O. G. Smolyanov, “Analytic properties of inifinite-dimensional distributions,” Usp. Mat. Nauk 45, No.3, 3–83 (1990).

    Google Scholar 

  5. 5.

    N. Bourbaki, Integration, Livre VI, Fasc. XIII, XXI, XXIX, XXXV. Ch. 1–9, Hermann, Paris (1965, 1967, 1963, 1969).

    Google Scholar 

  6. 6.

    J. P. R. Christensen, Topology and Borel Structure, North-Holland Math. Studies, 10, Elsevier, Amsterdam (1974).

    Google Scholar 

  7. 7.

    C. Constantinescu, Spaces of Measures, Springer, Berlin (1984).

    Google Scholar 

  8. 8.

    Yu. L. Dalecky and S. V. Fomin, Measures and Differential Equations in Infinite-Dimensional Spaces, Kluwer Acad. Publ., Dordrecht (1991).

    Google Scholar 

  9. 9.

    G. S. Djordjevic and B. Dragovich, “p-Adic and adelic harmonic oscillator with a time-dependent frequency,” Theor. Math. Phys., 124, No.2, 1059–1067 (2000).

    Google Scholar 

  10. 10.

    R. Engelking, General Topology [in Russian], Mir, Moscow (1986).

    Google Scholar 

  11. 11.

    H. Federer, Geometric Measure Theory, Springer, Berlin (1969).

    Google Scholar 

  12. 12.

    F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  13. 13.

    I. M. Gelfand and N. Ya. Vilenkin, Some Applications of Harmonic Analysis. Generalized functions [in Russian], 4, Fiz.-Mat. Lit., Moscow (1961).

    Google Scholar 

  14. 14.

    P. L. Henneken and A. Torta, Theory of Probability and Some of Its Applications [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  15. 15.

    E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, 2nd. ed., Springer, Berlin (1979).

    Google Scholar 

  16. 16.

    C. J. Isham and J. Milnor, in: Relativity, Groups and Topology. II, Editors B. S. De Witt, R. Stora, Elsevier, Amsterdam (1984), pp. 1007–1290.

    Google Scholar 

  17. 17.

    Y. Jang, Non-Archimedean quantum mechanics, Tohoku Math. Publ. 10 (1998).

  18. 18.

    A. Yu. Khrennikov, “Mathematical methods of non-Archimedean physics,” Usp. Mat. Nauk, 45, No.4, 79–110 (1990).

    Google Scholar 

  19. 19.

    A. Yu. Khrennikov and M. Endo, “Non-boundedness of p-adic Gaussian distributions,” Izv. Akad. Nauk SSSR, Ser. Mat., 56, 1104–1115 (1992).

    Google Scholar 

  20. 20.

    S. V. Ludkovsky, “Measures on groups of diffeomorphisms of non-Archimedean Banach manifolds,” Usp. Mat. Nauk, 51, No.2, 169–170 (1996).

    Google Scholar 

  21. 21.

    S. V. Ludkovsky, “Non-Archimedean polyhedral expansions of ultrauniform spaces,” Fund. Prikl. Mat., 6, No.2, 455–475 (2000).

    Google Scholar 

  22. 22.

    S. V. Ludkovsky, “Measures on diffeomorphism groups of non-Archimedean manifolds, representations of groups and their applications,” Theor. Math. Phys., 119, No.3, 381–396 (1999).

    Google Scholar 

  23. 23.

    S. V. Ludkovsky, “Quasi-invariant measures on non-Archimedean semigroups of loops,” Usp. Mat. Nauk, 53, No.3, 203–204 (1998).

    Google Scholar 

  24. 24.

    S. V. Ludkovsky, “Irreducible unitary representations of non-Archimedean groups of diffeomorphisms,” Southeast Asian Bull. Math., 22, 419–436 (1998).

    Google Scholar 

  25. 25.

    S. V. Ludkovsky, “Properties of quasi-invariant measures on topological groups and associated algebras,” Annales Mathem. B. Pascal, 6, No.1, 33–45 (1999).

    Google Scholar 

  26. 26.

    S. V. Ludkovsky, “Quasi-invariant measures on non-Archimedean groups and semigroups of loops and paths, their representations,” Annales Mathem. B. Pascal, 7, No.2, 19–53, 55–80 (2000).

    Google Scholar 

  27. 27.

    S. V. Ludkovsky, “Non-Archimedean free Banach spaces,” Fund. Prikl. Mat., 1, No.3, 979–987 (1995).

    Google Scholar 

  28. 28.

    S. V. Ludkovsky, “Quasi-invariant measures on a group of diffeomorphisms of an infinite-dimensional real manifold and induced irreducible unitary representations,” Rendiconti dell’ Istituto di Matem. dell’ Universita di Trieste. Nuova Serie, 31, 101–134 (1999).

    Google Scholar 

  29. 29.

    A. Madrecki, “Minlos’ theorem in non-Archimedean locally convex spaces,” Comment. Math. (Warsaw), 30, 101–111 (1991).

    Google Scholar 

  30. 30.

    A. Madrecki, “Some negative results on existence of Sazonov topology in l-adic Frechet spaces,” Arch. Math., 56, 601–610 (1991).

    Article  Google Scholar 

  31. 31.

    A. Madrecki, “On Sazonov type topology in p-adic Banach space,” Math. Z., 188, 225–236 (1985).

    Article  Google Scholar 

  32. 32.

    A. P. Monna and T. A. Springer, “Integration non-archimedienne,” Indag. Math., 25, 634–653 (1963).

    Google Scholar 

  33. 33.

    L. Narici and E. Beckenstein, Topological Vector Spaces, Marcel Dekker, New York (1985).

    Google Scholar 

  34. 34.

    A. C. M. van Rooij, Non-Archimedean Functional Analysis, Marcel Dekker, New York (1978).

    Google Scholar 

  35. 35.

    W. H. Schikhof, Ultrametric Calculus, Cambridge Univ. Press, Cambridge (1984).

    Google Scholar 

  36. 36.

    W. H. Schikhov, On p-Adic Compact Operators, Report No. 8911, Dept. Math. Cath. Univ., Nijmegen, the Netherlands (1989).

    Google Scholar 

  37. 37.

    W. H. Schikhof, “A Radon-Nikodym theorem for non-Archimedean integrals and absolutely continuous measures on groups,” Indag. Math. Ser. A, 33, No.1, 78–85 (1971).

    Google Scholar 

  38. 38.

    A. V. Skorohod, Integration in Hilbert Space, Springer, Berlin (1974).

    Google Scholar 

  39. 39.

    O. G. Smolyanov and S. V. Fomin, “Measures on linear topological spaces,” Usp. Mat. Nauk, 31, No.4, 3–56 (1976).

    Google Scholar 

  40. 40.

    F. Topsoe, “Compactness and tightness in a space of measures with the topology of weak convergence,” Math. Scand., 34, 187–210 (1974).

    Google Scholar 

  41. 41.

    F. Topsoe, “Some special results on convergent sequences of Radon measures,” Manuscripta Math., 19, 1–14 (1976).

    Article  Google Scholar 

  42. 42.

    N. N. Vahaniya, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions in Banach Spaces [in Russian], Nauka, Moscow (1985).

    Google Scholar 

  43. 43.

    V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics [in Russian], Nauka, Moscow (1994).

    Google Scholar 

  44. 44.

    A. Weil, Basic Number Theory, Springer, Berlin (1973).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

__________

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 149–199, 2003.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ludkovsky, S.V. Quasi-Invariant and Pseudo-Differentiable Measures with Values in Non-Archimedean Fields on a Non-Archimedean Banach Space. J Math Sci 128, 3428–3460 (2005). https://doi.org/10.1007/s10958-005-0280-2

Download citation

Keywords

  • Banach Space
  • Characteristic Functional
  • Infinite Product
  • Infinite Field
  • Kakutani Theorem