The universal completion of C(X) and unbounded order convergence

The universal completion of the Archimedean Riesz space C(X) of continuous, real valued functions on a completely regular space X is characterised as the space NL(X) of nearly finite, normal lower semi-continuous functions on X. As an application, we obtain, under additional assumptions on X, a characterisation of unbounded order convergence in C(X) as pointwise convergence everywhere except possibly on a set of first Baire category. This result is analogous to the situation in spaces of (real) p-summable functions, the sets of first Baire category now playing the role of null sets. We pursue this analogy further. First it is shown that, for a Baire space X, NL(X) is Riesz and algebra isomorphic to the space of real Borel measurable functions on X, with identification of functions differing at most on a set of first category. Secondly, through the use of density topologies and category measures, the extent to which our results can be cast in a measure-theoretic setting, and vice versa, is explored. Finally, through an application of the Maeda-Ogasawara Representation Theorem, we obtain a characterisation of those completely regular spaces X and Z such that C(X) and C(Z) have Riesz isomorphic universal completions.