A Characterization of the Vector Lattice 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 $1$ 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 $(X,\Sigma ,\mu )$.

13 pp