In [11] we have pointed out the importance of basic orthogonally scattered measures in the study of analytic and stochastic problems involving Hilbert spaces in which discreteness is absent and no orthogonal basis appears naturally. We also announced that such basic measures provide a natural foundation on which to build the Fourier analysis of L 2 functions over any locally compact abelian (l.c.a.) group X, and in [11’, §10] we gave this construction for the case X=R. In the present paper we shall carry over this construction for any l.c.a. group X.


Haar Measure Character Group Dual Group Compact Abelian Group Cardinality Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. S. Besicovitch, A general from of the covering principle and relative differentiation of additive functions, II. Proc. Cambridge Philos. Soc. 42 (1946), 1–10.CrossRefGoogle Scholar
  2. [2]
    S. Bochner, Inversion formulae and unitary transformations. Ann. of Math. 35 (1934), 111–115.CrossRefGoogle Scholar
  3. [3]
    S. Bochner, Harmonic analysis and the theory of probability. Univ. of Calif. Press, Berkeley, Calif., 1955.Google Scholar
  4. [4]
    H. Cartan & R. Godement, Théorie de la dualité et analyse harmonique dann les groupes abeliens localement compacts. Ann. Sci. École Norm. Sup. 64 (1947), 79–99.Google Scholar
  5. [5]
    G. Choquet, Les cônes convexes faiblement complets dans l’Analyse. Proc. Intern. Congr. Mathematicians, Stockholm (1962), 317–330.Google Scholar
  6. [6]
    J. L. Doob, Stochastic processes. Wiley, New York 1953.Google Scholar
  7. [7]
    R. E. Edwards & E. Hewitt, Pointwise limits for sequences of convolution operators. Acta Math. 113 (1965), 181–218.CrossRefGoogle Scholar
  8. [8]
    I. Gelfand, D. Raikov & G. Shilov, Commutative Normed Rings, Chelsea, New York 1964.Google Scholar
  9. [9]
    P. R. Halmos, Measure theory. Van Nostrand, New York 1950.Google Scholar
  10. [10]
    L. H. Loomis, An introduction to abstract harmonic analysis. Van Nostrand, New York 1953.Google Scholar
  11. [10]
    P. Masani, Orthogonally scattered measures. MRC Technical Report 738, 1967.Google Scholar
  12. [11]
    P. Masani, Orthogonally scattered measures. Advances in Math. 2 (1968), 61–117.CrossRefGoogle Scholar
  13. [12]
    M. A. Naimark, Normed rings. Noordhoff, Groningen 1964.Google Scholar
  14. [13]
    R.E.A.C. Paley & N. Wiener, Fourier transforms in the complex domain. Amer. Math. Soc, Providence, R. I., 1934.Google Scholar
  15. [14]
    L. S. Pontryagin, Topological groups (2nd Ed.). Gordan & Brown, New York 1966.Google Scholar
  16. [15]
    W. Rudin, Fourier analysis on groups. Interscience, New York 1962.Google Scholar
  17. [16]
    A. Weil, L’intégration dans les groupes topologiques et ses applications. Hermann, Paris 1940.Google Scholar
  18. [17]
    N. Wiener, Differential space, J. Math. & Phys. 2 (1923), 131–174.Google Scholar
  19. [18]
    N. Wiener, The Fourier integral and certain of its applications. Cambridge Univ. Press, Cambridge 1933.Google Scholar

Copyright information

© Springer Basel AG 1969

Authors and Affiliations

  • P. Masani
    • 1
  1. 1.Mathematics DepartmentIndiana UniversityUSA

Personalised recommendations