Given a probability measure space (X,Σ,μ)\documentclass[12pt]{minimal}
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\begin{document}$$(X,\Sigma ,\mu )$$\end{document}, it is well known that the Riesz space L0(μ)\documentclass[12pt]{minimal}
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\begin{document}$$L^0(\mu )$$\end{document} of equivalence classes of measurable functions f:X→R\documentclass[12pt]{minimal}
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\begin{document}$$f: X \rightarrow \mathbf {R}$$\end{document} is universally complete and the constant function 1\documentclass[12pt]{minimal}
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\begin{document}$$\varvec{1}$$\end{document} is a weak order unit. Moreover, the linear functional L∞(μ)→R\documentclass[12pt]{minimal}
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\begin{document}$$L^\infty (\mu )\rightarrow \mathbf {R}$$\end{document} defined by f↦∫fdμ\documentclass[12pt]{minimal}
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\begin{document}$$f \mapsto \int f\,\mathrm {d}\mu $$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$e>0$$\end{document} which admits a strictly positive order continuous linear functional on the principal ideal generated by e is lattice isomorphic onto L0(μ)\documentclass[12pt]{minimal}
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\begin{document}$$L^0(\mu )$$\end{document}, for some probability measure space (X,Σ,μ)\documentclass[12pt]{minimal}
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\begin{document}$$(X,\Sigma ,\mu )$$\end{document}.
