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Convex cones in analysis. Postface by Gustave Choquet. (Cônes convexes en analyse. Postface de Gustave Choquet.) (French) Zbl 0942.46002

Travaux en Cours. 59. Paris: Hermann. 245 p. (1999).
The book is devoted to the Choquet theory of integral representations, and the detailed study of weakly complete cones, in locally convex separated topological linear spaces. Gustave Choquet himself contributes a ‘Postface’ in which he briefly recounts the steps which led him to identify the class \({\mathcal S}\) of weakly complete strict cones and the notions of a cap of a cone and a conical measure which play a central role in the theory.
The first two chapters introduce this theory and it is applied in Chapter 3 to a variety of problems in classical analysis: proofs are given of the classical Bochner-Weil and Bernstein theorems and separate sections are devoted to the axiomatic potential theories of Brelot and Brauer, the inequation \(\mu*\sigma\leq \mu\) and the theorem of Choquet and Deny, Talagrand’s results concerning invariant measures and capacities, hypoelliptic operators, and the theory of quasi-invariant measures on \({\mathcal C}(\mathbb{R})\) after G. Royer and M. Yor.
Chapter 4 is devoted to Le Cam’s exposition, using conical measures, of statistical decision theory.
Chapter 5 establishes a canonical relation between conical measures, functions of negative type and the class of mathematically beautiful objects, the zonoforms, generalizations of Coxeter’s zonohedra.
Chapters 6 and 7 discuss the relation between conical measures and cylindrical measures, and the class of bireticulated cones respectively.
The final Chapter 8 presents work, mainly of the author, concerning the relation between weakly complete cones in a Banach space and normal cones in the space.
The book begins with an elementary introduction of six unnumbered pages, there is a detailed historical appendix and throughout the text there are careful attributions. The result is a rich and readable book of considerable charm, the short sections of which establish a pressing tempo.

MSC:

46A55 Convex sets in topological linear spaces; Choquet theory
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46G12 Measures and integration on abstract linear spaces
46A40 Ordered topological linear spaces, vector lattices
28B05 Vector-valued set functions, measures and integrals
28B15 Set functions, measures and integrals with values in ordered spaces
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
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