On multiparameter ergodic and martingale theorems in infinite measure spaces (Q1077067)

In Z. Wahrscheinlichkeitstheor. Verw. Geb. 63, 43-49 (1983; Zbl 0489.46021) the second author obtained a unified and remarkably simple approach to many multiparameter pointwise ergodic theorems and martingale theorems. The method is now extended to infinite measure spaces (\(\Omega\),\({\mathfrak A},\mu)\). The proper setting is now given by the function spaces \(R_ k\) introduced by \textit{N. A. Fava} [Stud. Math. 42, 271-288 (1972; Zbl 0237.47006)]. For \(k=1,2,...\), \(R_ k\) is the class of functions f with \[ \int | nf| (\log^+| nf|)^ k d\mu <\infty \quad for\quad all\quad n\geq 1. \] \(R_ 0\) is the class of f with \(\int | f| 1_{\{| f| >1+n\}}d\mu <\infty\) for all n. \(R_ 0\) is shown to be an order continuous Banach lattice for the \(L_ 1+L_{\infty}\)-norm \(\| f\| =\inf \{\| g\|_ 1+\| h\|_{\infty}:\) \(f=g+h\}\). To start an inductive argument the authors prove a martingale theorem and the Dunford-Schwartz ergodic theorem for functions in \(R_ 0\) and Rotas theorem in \(R_ 1\). This leads to multiparameter convergence theorems in \(R_ k\).