Generalization of Eberlein's and Sine's ergodic theorems to $LR$-nets

The notion of $LR$-nets provides an appropriate setting for study of various ergodic theorems in Banach spaces. In the present paper, we prove Theorems 2.1, 3.1 which extend Eberlein's and Sine's ergodic theorems to $LR$-nets. Together with Theorem 1.1, these two theorems form the necessary background for further investigation of strongly convergent $LR$-nets. Theorem 2.1 is due to F. Räbiger, and was announced without a proof in [1].