On the Ergodic Theorem of E. Hopf

In the paper we extend the ergodic theorem of E. Hopf to a large class of Archimedean Riesz spaces. In order to obtain the extension, we define and study a new type of convergence in Archimedean Riesz spaces. The new type of convergence (which we call individual convergence) was inspired by a work of Nakano (Ann. Math.49 (1948), 538-556) (from which we also borrowed its name) and by a paper of Ornstein (in "Advances in Probability and Related Topics" (Peter Ney, Ed.), Vol. 2, pp. 85-115, Dekker, New York, 1970) on modifications and the ratio ergodic theorem.