Topologies and projections on Riesz spaces

Abstract

Introduction. Among the various questions about topologies on Riesz spaces, there is a particularly important one which arises out of Daniell's work on integration. The present paper will be centered around this question, but will not be confined to it. Daniell [S] considers a space V whose elements are real-valued functions on an abstract set E and which is closed under the natural linear and lattice operations. He starts with a positive linear functional F on V (that is, Ff _0 whenever f(x) > 0 for all xGE) endowed with the property that fn(x) Tf(x) pointwise implies Ffn-*Ff. Daniell extends F from V to a larger class of functions in such a way that the extended F satisfies the theorem of Lebesgue: Under suitable conditions of boundedness, if fn-*f pointwise and the Ff. are defined, Ff is defined and Ffn -Ff. It follows that the extended F is the integral corresponding to a measure on E. In accordance with the work of Daniell, we call a positive linear functional F on V an integral if it has the property: fn Tf pointwise implies