Stonean Lattices, Measures and Completeness

Abstract

If L = linear space and lattice of real valued functions on some set X and φ:L→reals is linear, for φ to be representable in the form φ = ∫··dμ with some finitely or σ-additive μ, certain continuity conditions for φ are necessary and sufficient [6], for example Daniell’s condition in the σ-additive case. If X is compact, L contains all continuous functions f and φ ≥ O, Daniell’s condition is automatically fulfilled because of Dini’s theorem and 1∈L, yielding Riesz’s theorem with Baire measures. This is true if X is any topological space, L containing all continuous f with compact support or vanishing at ∞, and, somewhat unexpectedly, if L contains all continuous f (p.e.[7]). Similarly, if L contains all bounded f on any topological X, all linear φ ≥ O are integrals with some finitely additive μ [7].