On the Ergodic Theorem of E. Hopf

On the Nakano Individual Convergence

“…By Proposition 2 of [9), Bos((ufl)flE,v) and BNos((u)flE,r) are projection bands in E, and E is the order direct sum of Bos((u fl ) flE tq) and BNos((u)eJ). As in [9), let Ba(( u )€,v) be the projection band in E generated by BoO((un)nEflv)UB0s((un)nEIv).…”

“…Let (ufl)nEJp., be a sequence of elements of E. and let u E E. We say that the sequence Now let (u)€jN be a sequence of elements of Esuch that u, > 0 for every n E IN. As in [9], let B,((u fl ) fl jp,,) he the largest hand in E on which (un)njpj is unbounded. Set (see [9]) B((n)n€v) = u = 0. or u 54 0 and, for every v E E with 0 < v < Jul, v 0 there exist u E wEE with 0<w<v,w0 and a,/3EJR with 0 <fi< a such that…”

“…Unless explicitly stated otherwise, the terminology used in this paper can be found in the books of Aliprantis and Burkinshaw [1], Luxemburg and Zaanen [5], Schaefer [7], and in our papers [8,9].…”

“…In the next section we will recall several notions and results of [6] and [9] .which will be needed throughout the paper; thus, we will review the definitions of the Nakano R. Zaharopol: Bingham . Univ., Dep.…”

“…Univ., Dep. Math., Binghamton, NY 13902-6000, USA ISSN 0232-2064 / $ 2.50 © Heldermann Verlag Berlin individual convergence and the Nakano individual limit in the more general setting of Archimedean Riesz spaces (rather than countably order complete Riesz spaces in which Nakano originally stated his definitions), and we will describe briefly the notion of individual convergence of [9], as well as the results which make our definition possible. In Section 3 we will define the individual limit of a sequence ( u ' )^E BV of elements of an Archimedean Riesz space, and we will show that if there exists an individual limit of (un) E p.J, theii converges individually.…”