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(μ) of equivalence classes of measurable functions f: X→ R is universally complete and the constant function 1 is a weak order unit. Moreover, the linear functional L∞(μ) → R defined by f↦∫fdμ 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(μ) , for some probability measure space (X, Σ , μ).