Two general types of pointwise ergodic theorems have been studied: those as t approaches infinity, and those as t approaches zero. This paper deals with the latter case, which is referred to as the local case.
Let (X, , μ) be a complete, σ-finite measure space. Let {Tt
} be a strongly continuous one-parameter semi-group of contractions on , defined for t ≧ 0. For Tt
positive, it was shown independently in [2] and [5] that
1.1
almost everywhere on X, for any f ∊ L1. The same result was obtained in [1], with the continuity assumption weakened to having it hold for t > 0.