Banach's principle in vector lattices

Abstract

We develop an abstract Banach principle in vector lattices and apply it to obtain lattice-theoretic versions of the individual and maximal ergodic theorems, without recourse to any measure representation. Considering a sequence of bounded operators with values in a Dedekind $\sigma$-complete vector lattice endowed with a locally solid topology satisfying the $\sigma$-Lebesgue property, we prove that the set of points at which the sequence is order convergent is a closed subspace and coincides with the whole space whenever convergence holds on a dense subset. Investigating positive, power-bounded, mean ergodic operators on order continuous Banach lattices, we construct a topology on the universal completion induced by a strictly positive order continuous functional and prove that the Ces\`aro means converge in order in the universal completion and, in particular, uo-converge in the original lattice. Moreover, we introduce the notion of a superinvariant pair and derive a lattice-theoretic Hopf inequality together with a weak type estimate via band projections, which yields an abstract maximal ergodic theorem. A spectral-theoretic version of the theorem follows from classical Perron--Frobenius theory. Finally, we specialise the abstract framework to the model space $L^0(\Omega)$ and revisit the classical Banach principle, the Hopf--Dunford--Schwartz theorem and Doob's martingale convergence theorem.