A new approach is developed in
the theory of pointwise ergodic theorems. Our consideration is based upon
Ωμp(0 ≦ p < ∞), which is a linear space containing properly the linear span of
Up>1Lp(X;𝒳), where (X,ℱ,μ) is a σ-finite measure space and 𝒳 is a reflexive
Banach space. Some weak and strong type inequalities are proved as vector valued
generalizations of the Dunford and Schwartz’s results, and then, used to
study the integrability of the ergodic maximal function. These results do
make it possible to extend the Chacon’s vector valued ergodic theorem.
We have analogous extensions for the case of continuous semigroups, and
the local ergodic theorem is shown to hold on Ωμ0. The results include two
applications to the random ergodic theorem and the “strong differentiability”
theorem.
