A Characterization of the Vector Lattice of Measurable Functions

Abstract

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 \(f: X \rightarrow \mathbf {R}\) is universally complete and the constant function \(\varvec{1}\) is a weak order unit. Moreover, the linear functional \(L^\infty (\mu )\rightarrow \mathbf {R}\) 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 \((X,\Sigma ,\mu )\).