  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 $(X,\Sigma,\mu)$.
