Riesz spaces

The Riesz Representation is of great importance in analysis. But in certain situations such as in infinite dimensional Hilbert spaces it fails to apply. Here we present an extensive proof of a relevant generalization using sigma topology as discovered by Edwin Beggs.

Locally compact spaces having property (B) are introduced. (B) is the property that each bounded subset of the topological vector space K(X) is order bounded. It is shown that there exist spaces X exhibiting (B) that are not paracompact. Also discussed are the consequences of this property for the dual and bidual of K(X) Both show features resembling those from the Banach space case (X compact). Under (B), for example, the bidual of K(X) is Riesz isomorphic to K(Y) for a suitable locally compact space Y. We also introduce doubly bounded Borel functions Bb(X) which are bounded and vanish outside some compact set. Without using the Riesz representation theorem they are imbedded in the bidual of K(X), extending the approach of H. H. Schaefer (19) (20) for compact spaces to locally compact spaces. Further it is shown that the regular Borel measures form a band in the Riesz dual of Bb(X) These results permit to give a topological characterization of regular Borel measures on X, which yields the Riesz representation theorem as well as a distinguishing property of regular Borel measures as fairly immediate consequences. Finally, some relations between Baire and Borel measures are discussed.