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. [ABSTRACT FROM AUTHOR]