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E. Hopf’s Theorem

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

Abstract

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.

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© Hindustan Book Agency 2013

Authors and Affiliations

  1. 1.University of MumbaiIndia

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