Let X be a reflexive Banach space and (Ω,Σ,μ) be a σ-finite measure space. Let d ≥ 1 be an integer and T=T(u):u=($u_1$, ... ,$u_d)$, $u_i$ ≥ 0, 1 ≤ i ≤ d be a strongly measurable d-parameter semigroup of linear contractions on $L_1$((Ω,Σ,μ);X). We assume that to each T(u) there corresponds a positive linear contraction P(u) defined on $L_1$((Ω,Σ,μ);ℝ) with the property that ∥ T(u)f(ω)∥ ≤ P(u)∥f(·)∥(ω) almost everywhere on Ω for all f ∈ $L_1$((Ω,Σ,μ);X). We then prove stochastic and pointwise ergodic theorems for a d-parameter...

Previously we obtained stochastic and pointwise ergodic theorems for a continuous d-parameter additive process F in L₁((Ω,Σ,μ);X), where X is a reflexive Banach space, under the condition that F is bounded. In this paper we improve the previous results by considering the weaker condition that the function $W\left(·\right)=esssup{\left|\right|F\left(I\right)\left(·\right)\left|\right|:I\subset [0,1)}^d$ is integrable on Ω.