E. Hopf’s Theorem

Part of the Texts and Readings in Mathematics book series (volume 6)


In this chapter we will discuss a necessary and sufficient condition for a Borel automorphism to admit a finite invariant measure. The necessary notion of incompressibility was already formulated by E. Hopf ([5], 1932). We will combine a refined form of this notion with certain observations of V. V. Srivatsa to give a measure free proof of the pointwise ergodic theorem. Application of Ramsay-Mackey theorem and some classical measure theory then provides us with an invariant probability measure when the space is incompressible. We will also briefly mention generalisations to Polish group actions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. D. Birkhoff and P. A. Smith. Structure Analysis of Surface Transformations, Journ. de. Math. Tome VII — Fasc IV, (1928), 345–379, Birkhoff: Collected Mathematical Papers, Vol. 2, 360–394, Dover,1968.zbMATHGoogle Scholar
  2. [2]
    P. Chaube and M. G. Nadkarni. On Orbit Equivalence of Borel Automorphisms, Proc. Indian Acad Sci. (Math. Sci.), 99 (1989), 255–261.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    S. Eigen, A. Hajian and M. G. Nadkarni. Weakly Wandering Sets and Compressibility in Descriptive Setting. Proc. Indian Acad. Sci. (Math. Sci.), 103 (1993), 321–327.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. B. Hajian and S. Kakutani. Weakly Wandering Sets and Invariant Measures, Trans. Amer. Math. Soc., 110 (1964), 136–151.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    E. Hopf. Theory of Measures and Invariant Integrals, Trans. Amer. Math. Soc., 34 (1932), 373–393.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    F. J. Murray and J. von Neumann. On Rings of Operators, Ann. Math., 37 (1936), 116–129.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. G. Nadkarni. A Descriptive Characterisation of Ergodic Systems, Sankhya, The Indian Journal of Statistics, 45 (1983), 271–287.zbMATHGoogle Scholar
  8. [8]
    M. G. Nadkarni. On The Existence of A Finite Invariant Measure, Proc. Indian Acad. Sci. (Math. Sci.) vol., 100 (1990), 203–220.MathSciNetzbMATHGoogle Scholar
  9. [9]
    S. Wagon. The Banach-Tarski Paradox, Cambridge University Press, 1986.zbMATHGoogle Scholar
  10. [10]
    P. Zakrzewski. The Existence of Invariant Probability Measures For A Group of Transformations, Israel Journal of Mathematics 83 (3), (1993), 343–352.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hindustan Book Agency 2013

Authors and Affiliations

  1. 1.University of MumbaiIndia

Personalised recommendations