Let (X,Σ,μ) be a σ-finite
measure space and Lp(X,Σ,μ), 1 ≦ p ≦∞, the usual Banach spaces of complex
valued functions. Let {Tt: t ≧ 0} be a strongly continuous semigroup of contractions
of Lp(X,Σ,μ) for some 1 ≦ p < ∞ and set Rλf =∫0∞e−λtTtf dt, λ > 0.
If ∥Tt∥∞≦ 1 for all t ≧ 0, then limλ→∞λRλf(x) = f(x) a.e. for all
f ∈ Lp(X,Σ,μ).