Let (X,ℱ,μ) be a σ-finite
measure space and let L1 denote the usual Banach space of equivalence classes of real
valued integrable functions on X. We shall not distinguish between the equivalence
classes and the functions themselves. Relations between functions are assumed to
hold in an a.e. sense.
Throughout this paper {Tt}t>0 will denote a strongly continuous semigroup of
linear contractions on L1. That is:
each Tt is a linear operator on L1, with norm not more than 1,
Ts+t= TtTs for all t,s > 0,
for all f ∈ L1 and t > 0, lims→t,s>0;∥Tsf − Ttf∥ = 0.
We will prove the pointwise local ergodic theorem for such a semi
group.
