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Compact convex sets and boundary integrals…
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Compact convex sets and boundary integrals

by Erik M. Alfsen

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Indeholder "1. Preface", "Chapter 1. Representations of Points by Boundary Measures", " Paragraph 1. Distinguished Classes of Functions on a Compact Convex Set", " Classes of continuous and semicontinuous, affine and convex functions", " Uniform and pointwise approximation theorems", " Envelopes", " Grothendieck's completeness theorem", " Theorems of Banach-Dieudonne and Krein-Smulyan", " Paragraph 2. Weak Integrals, Moments and Barycenters", " Preliminaries and notations from integration theory", " An existence theorem for weak integrals", " Vague density of point-measures with prescribed barycenter", " Choquet's barycenter formula for affine Baire functions of first class, and a counterexample for affine functions of higher class", " Paragraph 3. Comparison of Measures on a Compact Convex Set", " Ordering of measures", " The concept of dilation for simple measures", " The fundamental lemma on the existence of majorants", " Characterization of envelopes by integrals", " Dilation of general measures", " Cartier's Theorem", " Paragraph 4. Choquet's Theorem", " A characterization of extreme points by means of envelopes", " The concept of a boundary set", " Herve's theorem on the existence of a strictly convex function on a metrizable compact convex set", " The concept of a boundary measure, and Mokobodzki's characterization of boundary measures", " The integral representation theorem of Choquet and Bishop - de Leeuw", " A maximum principle for superior limits of 1.s.c. convex functions", " Bishop - de Leeuw's integral theorem relatively to a σ-field on the extreme boundary", " A counterexample based on the 'porcupine topology'", " Paragraph 5. Abstract Boundaries Defined by Cones of Functions", " The concept of a Choquet boundary", " Bauer's maximum principle", " The Choquet-Edwards theorem that Choquet boundaries are Baire spaces", " The concept of a Silov boundary", " Integral representation by means of measures on the Choquet boundary", " Paragraph 6. Unilateral Representation Theorems with Application to Simplicial Boundary Measures", " Ordered convex compacts", " Existence of maximal extreme points", " Characterization of the set of maximal extreme points as a Choquet boundary", " Definition and basic properties of simplicial measures", " Existence of simplicial boundary measures, and the Caratheodory Theorem in R^n", " Decomposition of representing boundary measures into simplicial components", "Chapter 2. Structure of Compact Convex Sets", " Paragraph 1. Order-unit and Base-norm Spaces", " Basic properties of (Archimedean) order-unit spaces", " A representation theorem of Kadison", " The vector-lattice theorem of Stone-Kakutani-Krein-Yosida", " Duality of order-unit and base-norm spaces", " Paragraph 2. Elementary Embedding Theorems", " Representation of a closed subspace A of C_R(X) as an A(K)-space by the canonical embedding of X in A*", " The concept of an 'abstract compact convex' and its regular embedding in a locally convex Hausdorff space", " The connection between compact convex sets and locally compact cones", " Paragraph 3. Choquet Simplexes", " Riesz' decomposition property and lattice cones", " Choquet's uniqueness theorem", " Choquet-Meyer's characterizations of simplexes by envelopes", " Edward's separation theorem", " Continuous affine extensions of functions defined on compact subsets of the extreme boundary of a simplex", " Affine Borel extensions of functions defined on the extreme boundary of a simplex", " Examples of 'non-metrizable' pathologies in simplexes", " Paragraph 4. Bauer Simplexes and the Dirichlet Problem of the Extreme Boundary", " Bauer's characterizations of simplexes with closed extreme boundary", " The Dirichlet problem of the extreme boundary", " A criterion for the existence of continuous affine extensions of maps defined on extreme boundaries", " Paragraph 5. Order Ideals, Faces, and Parts", " Elementary properties of order ideals and faces", " Extension property and characteristic number", " Archimedean and strongly Archimedean ideals and faces", " Exposed and relatively exposed faces", " Specialization to simplexes", " The concept of a 'part', and an inequality of Harnack type", " Characterization of the parts of a simplex in terms of representing measures", " An example of an Archimedean face which is not strongly Archimedean", " Paragraph - 6. Split-faces and Facial Topology", " Definition and elementary properties of split faces", " Characterization of split faces by relativization of orthogonal measures", " An extension theorem for continuous affine functions defined on a split face", " The facial topology", " Specialization to simplexes", " Near-lattice ideals, and primitive ideal space", " The connection between facial topology and hull kernel topology", " Compact convex sets with sufficiently many inner automorphisms", " A remark on the applications to C*-algebras", " Paragraph 7. The Concept of Center for A(K)", " Extension of facially continuous functions", " The facial topology is Hausdorff for Bauer simplexes only", " The concept of center, and the connections with facially continuous functions and order-bounded operators", " Convex compact sets with trivial center", " An example of a prime simplex", " Stormer's characterization of Bauer simplexes", " Paragraph 8. Existence and Uniqueness of Maximal Central Measures Representing Points of an Arbitrary Compact Convex Set", " The relation x o y, and the concept of a primary point", " A point x is primary iff the local center at x is trivial", " The concept of a central measure", " Existence and uniqueness of maximal central measures in a special case", " The 'lifting' technique", " Wils' theorem on the existence and uniqueness of maximal central measures which are pseudo-carried by the set of primary points", "Appendix", "References", "Subject Index".

Hmm, jeg kan vist ikke engang få hul på det indledende kapitel uden hjælp. ( )
  bnielsen | Dec 26, 2017 |
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