Unbounded norm topology beyond normed lattices

TL;DR

This paper extends the concept of unbounded norm convergence from normed lattices to more general settings, including universal completions and spaces of measurable functions, unifying various modes of convergence.

Contribution

It introduces a generalized un-convergence framework beyond normed lattices, encompassing spaces like $L_0(u)$ and coordinate-wise convergence, with several extended results.

Findings

Un-convergence in $L_0(u)$ coincides with convergence in measure.

Un-convergence on $u$ with atomic, order complete $X$ matches coordinate-wise convergence.

Several classical un-convergence results are extended to the new setting.

Abstract

In this paper, we generalize the concept of unbounded norm (un) convergence: let $X$ be a normed lattice and $Y$ a vector lattice such that $X$ is an order dense ideal in $Y$; we say that a net $(y_\alpha)$ un-converges to $y$ in $Y$ with respect to $X$ if $\Bigl\lVert\lvert y_\alpha-y\rvert \wedge x\Bigr\rVert\to 0$ for every $x\in X_+$. We extend several known results about un-convergence and un-topology to this new setting. We consider the special case when $Y$ is the universal completion of $X$. If $Y=L_0(\mu)$, the space of all $\mu$-measurable functions, and $X$ is an order continuous Banach function space in $Y$, then the un-convergence on $Y$ agrees with the convergence in measure. If $X$ is atomic and order complete and $Y=\mathbb R^A$ then the un-convergence on $Y$ agrees with the coordinate-wise convergence.