Ergodic theory and the strong law of large numbers on Riesz spaces
The authors continue to develop Riesz space analogues of some central notions and results of probability theory. To give a flavour of the general set-up, note that the space \( L^1({\mathcal F}) \) of all integrable random variables on some probability space \( (\Omega,{\mathcal F},P) \) is not only a Banach space, but also a vector lattice (=: Riesz space) containing the constants, and that the conditional expectation \( E^{{\mathcal G}} : L^1({\mathcal F}) \to L^1({\mathcal F}) \) with respect to some sub-\(\sigma\)-algebra \( {\mathcal G} \subseteq {\mathcal F} \) is a linear operator which is not only norm continuous, but also positive, order continuous and idempotent with \( E^{{\mathcal G}}(L^1({\mathcal F})) = L^1({\mathcal G}) \) and that it preserves the constants. Dropping the probability space, the authors consider a general Riesz space \( E \) with a weak order unit and define a conditional expectation \( T : E \to E \) to be a positive order continuous projection for which \( T(E) \) is a Dedekind complete Riesz subspace of \( E \) and \( Te \) is a weak order unit whenever \( e \in E \) is a weak order unit. What is even more striking is the new notion of independence of Riesz subspaces with respect to a conditional expectation operator \( T : E \to E \), which corresponds perfectly to the notion of conditional independence of sub-\(\sigma\)-algebras with respect to a sub-\(\sigma\)-algebra \( {\mathcal G} \subseteq {\mathcal F} \). In this general setting, the authors prove abstract versions of ergodic theorems, of Kolmogorov's Zero-One Law, and of the Strong Law of Large Numbers for sequences of the form \( \frac{1}{n} \sum_{j=0}^{n-1} S^jf \) where \( f \in E \) and \( S : E \to E \) is an order continuous Riesz homomorphism.
scientific article; zbMATH DE number 5078936