Some Discussions Of Birkhoff Average

In the classic Birkhoff pointwise ergodic theorem,we know that:the natural number sequence N is a good universal sequence for pointwise convergence in the space of L1.That is for any f?L1 and measure preserving systems(X,B,?,T),we havethe main contents of this paper are the following:In chapter 2,we introduces the stability and sub-stability for the integer sequence{mk}in different Orlicz Spaces;In chapter 3,we discussed the almost everywhere convergence for the following eneralized Birkhoff averagee and give three criterions for the convergence of the above limit.In chapter 4,we mainly consider the almost everywhere convergence of weighted eneralized Birkhoff average,and discusses the following problems in turn:(i)under the condition that the standard Birkhoff average converges according to the norm(the weighted sequence is regarded as a disturbance),we consider the suuffcient condition that make the weighted sequence{wn}be a benign disturbance;(ii)the almost everywhere convergence of the generalized Birkhoff average weighted by Davenport and Gelfond type sequences.The main contents of this paper comes from the summary and extraction of the lecture I studied during my master's degree,which is more inclined to reading reports in nature.Specific literatures are reflected in the chapter review of each chapter.Some examples,corollaries and propositions in the paper are given by myself.