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A Characterization of the Vector Lattice of Measurable Functions (2203.07763v1)
Published 15 Mar 2022 in math.FA, math.CA, and math.PR
Abstract: Given a probability measure space $(X,\Sigma,\mu)$, it is well known that the Riesz space $L0(\mu)$ 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 $L0(\mu)$, for some probability measure space $(X,\Sigma,\mu)$.