In this chapter the basic objects studied in ergodic theory, measure-preserving transformations, are introduced. Some examples are given, and the relationship between various mixing properties is described. The mean and pointwise ergodic theorems are proved. An approach to the maximal ergodic theorem via a covering lemma is given, which will be extended in Chapter 8 to more general group actions.
KeywordsErgodic Theorem Compact Abelian Group Bernoulli Shift Ergodic Average Covering Lemma
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