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A Characterization of the Vector Lattice of Measurable Functions

Given a probability measure space $(X,Σ,μ)$, it is well known that the Riesz space $L^0(μ)$ 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 $L^\infty(μ)\to \mathbf{R}$ defined by $f \mapsto \int f\,\mathrm{d}μ$ 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 $L^0(μ)$, for some probability measure space $(X,Σ,μ)$.