Stonean Lattices, Measures and Completeness
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 , 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.). Similarly, if L contains all bounded f on any topological X, all linear φ ≥ O are integrals with some finitely additive μ .
Unable to display preview. Download preview PDF.
- Bauer, H., Sur l’équivalence des théories de l’intégration selon N. Bourbaki et selon M.H. Stone., Bull. Soc. Math. France 85 (1957), 51–75.Google Scholar
- Bichteler, K., Integration Theory. (Lecture Notes 315) Berlin/New York, Springer-Verlag 1973.Google Scholar
- Dunford, N. — Schwartz, J. T., Linear Operators. I. 4th ed., New York, Interscience Publ. 1967.Google Scholar
- Günzler, H., Linear functionals which are integrals. Rend. Sem. Mat. Fis. Milano 1974.Google Scholar
- Günzler, H., Integral representations on function lattices. Rend. Sem. Mat. Fis. Milano 1974/75.Google Scholar
- Markoff, A., On mean values and exterior densities.. Mat. Sbornik N.S.4, 46 (1938), 165–191.Google Scholar
- Pollard, D. — Topsøe, F., A unified approach to Riesz type representation theorems. To appear.Google Scholar
- Topsøe, F., Topology and Measure. (Lecture Notes 133) Berlin/New York, Springer-Verlag 1970.Google Scholar