Weighted Ergodic Theorem

In this thesis, we will mainly discuss the ergodic theorem and the weighted er-godic theorem. The main contents include: Firstly, we present two proofs of Birkhoff's pointwise ergodic theorem. One of the method is based on the space decomposition and maximal inequality; and the other one is based on the idea of nonstandard analy-sis. And we give an application of Birkhoff 's ergodic theorem: the frequency of the first digit of the sequence ?2ⁿ? ?Gelfand's problem?. Secondly,we make a brief introduction to the development of Wiener-Wintner ergodic theorem . We prove the theorem by Kronecker factor and Van der Corput equality. Lastly, we define Davenport weights and Davenport exponents. When a weight ?w_n??l^? satisfy the Davenport condition?see ?3.9??, the weighted ergodic averages ?1/n??_n=1^Nw_nf?Tⁿx? converge to zero almost everywhere for any preserving dynamic systems ?X,B,?,T?, any f?L¹???. This result is referred to as the weighted ergodic theorem with Davenport weights. The weighted ergodic theorem of M???bius weight is a special case.