Pointwise ergodic theorems for higher levels of mixing

We prove strengthenings of the Birkhoff Ergodic Theorem for weakly mixing and strongly measure preserving systems. show that our pointwise theorem systems is strictly stronger than Wiener-Wintner Theorem. also Theorems characterize respectively. The methods this paper allow one to an enhanced ergodic other levels hierarchy such as ergodicity mild but not K-mixing. author plans include these additional theorems in his thesis.