We consider a positive invertible Lamperti operator T with positive inverse. Our result concerns the averages Ak,n, 0≤k,n∈Z, and the ergodic maximal operatorMTf=supk,n≥0|Ak,nf|=supk,n≥0|1k+n+1∑j=−knTjf|, the ergodic Hilbert transform defined by HTf(x)=limn→∞∑j=1n1j(Tjf(x)−T−jf(x)) and the ergodic power functions defined by Pr,Tf(x)=(∑n=0∞|An+1,0f(x)−An,0f(x)|r+|A0,n+1f(x)−A0,nf(x)|r)1/r, for 1<r<+∞. Several authors proved that if the averages are uniformly bounded in Lp, 1<p<∞, then these operators are bounded in Lp[23,24,27,28,21]. Regarding the case p=1, Gillespie and Torrea [12] showed that this condition is not sufficient to assure MT is of weak type (1,1). We provide two sufficient conditions that recall the assumptions in the Dunford-Schwartz theorem: if the averages are uniformly bounded in L1 and in L∞ then MT, HT and Pr,T apply L1 into weak-L1 and the corresponding sequences of functions in L1 converge a.e. and in measure. Furthermore, we reach the same conclusions by assuming a weaker condition: we replace the uniform boundedness of the averages An,n in L∞ by the assumption that, for a fixed p∈(1,∞), the averages associated with a modified operator Tp, related with T, are uniformly bounded in Lp. We end the paper showing examples of nontrivial operators satisfying the assumptions of the two main results.