E. Hopf’s Theorem
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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 (, 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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