A Characterization of the Vector Lattice of Measurable Functions

TL;DR

This paper characterizes the structure of the vector lattice of measurable functions, showing that any universally complete Riesz space with certain properties is isomorphic to an $L^0$ space over some probability measure.

Contribution

It proves a converse characterization of $L^0( ext{measure space})$ as universally complete Riesz spaces with a weak order unit and a strictly positive order continuous linear functional.

Findings

Any such Riesz space is lattice isomorphic to some $L^0( ext{measure space})$.

The properties of $L^0( ext{measure space})$ are necessary and sufficient for this isomorphism.

The result links abstract Riesz space properties to concrete spaces of measurable functions.

Abstract

Given a probability measure space $(X,\Sigma,\mu)$, it is well known that the Riesz space $L^0(\mu)$ of equivalence classes of measurable functions $f: X \to \mathbf{R}$ is universally complete and the constant function $\mathbf{1}$ is a weak order unit. Moreover, the linear functional $L^\infty(\mu)\to \mathbf{R}$ defined by $f \mapsto \int f\,\mathrm{d}\mu$ is strictly positive and order continuous. Here we show, in particular, that the converse holds true, i.e., any universally complete Riesz space $E$ with a weak order unit $e>0$ which admits a strictly positive order continuous linear functional on the principal ideal generated by $e$ is lattice isomorphic onto $L^0(\mu)$, for some probability measure space $(X,\Sigma,\mu)$.