Strong completeness of Lp-type vector lattices

TL;DR

This paper proves the strong completeness of Lp-type vector lattices with a conditional expectation operator, extending previous results and unifying various convergence types, with applications to ergodic theory.

Contribution

It establishes the strong completeness of Lp(T) spaces for general p, resolving a long-standing open problem and unifying multiple convergence concepts in vector lattices.

Findings

Lp(T) spaces are strongly complete for all p, extending the p=2 case.

Unified framework for order, norm, and weak convergence.

New results in ergodicity for systems preserving conditional expectations.

Abstract

Let E be a Dedekind complete Riesz space with weak unit e, equipped with a conditional expectation operator T. We prove that the spaces Lp(T), with their natural vector-valued norms, are strongly complete, extending the p=2 case of Kuo, Kalauch, and Watson. This resolves a question that has remained open for several years. We begin by studying a general type of convergence and its unbounded modification, unifying and generalizing order, norm, and absolute weak convergence while providing simpler proofs. As an application, we consider vector-valued norms and their unbounded variants, generalizing strong convergence in Lp-spaces and convergence in probability. This framework establishes the completeness of Lp(T) and of the universal completion E^{u}, reinforcing the uo-completeness of universally complete vector lattices. Finally we apply our main theorem to obtain a new result in ergodicity for conditional preserving systems.