Bourgain-uo sequential completeness in vector lattices

Abstract

We revisit Bourgain's 1981 counterexample to the sequential completeness of the `pointwise plus domination' convergence on ${\ell }_1$ from the perspective of vector lattices. In this setting, we show that for sequences the associated notion of Bourgain--uo convergence coincides with ordinary order convergence. Motivated by Bourgain's construction, we introduce a strengthened, subsequence-invariant notion of Cauchy sequence: a sequence $(x_n)$ in a vector lattice $E$ is called Buo-Cauchy if for every strictly increasing sequence $(n_k)$ the differences $x_{n_{k+1}}-x_{n_k}$ converge to $0$ in order in $E$. We first show that sequential Buo-completeness forces $\ddot{I}ƒ$-order completeness. Thus every non-$\ddot{I}ƒ$-order complete vector lattice fails sequential \Buo-completeness. In particular, free Banach lattices $\mathrm{FBL}(E)$ are not sequentially Buo-complete whenever $\dim E>1$. On the positive side, we prove that the classical sequence lattices $c_0$ and ${\ell }_{\infty }$ are sequentially Buo-complete: every Buo-Cauchy sequence converges in order, and hence in the Buo sense. Finally, we obtain a sharp metric characterisation for bounded Lipschitz function lattices: the vector lattice ${\mathrm{Lip}}_b(X)$ of bounded Lipschitz functions on a metric space $(X,d)$ is sequentially Buo-complete if and only if $X$ is uniformly discrete.