Some Choquet theorems (Q1088111)

By demonstrating three particular theorems that give a new viewpoint of known theorems, the author introduces a type of geometric Choquet theorem. The first two are compact, the third is non-compact, and all three are existence and uniqueness theorems. The first theorem, theorem 3.1 (see below), is equivalent to an analytic Choquet theorem - namely, theorem 1.4. The second, theorem 3.2 (see below), is a sharpened version of theorem 3.1; its existence part is equivalent to the Choquet-Bishop-de Leeuw theorem. The third, theorem 3.4 (see below), is a non-compact version of theorem 3.2. Theorem 3.1. Let K be a compact absolutely convex subset of a real (complex) locally convex topological vector space, and let \(x\in K\). Then there exists a real (complex) boundary measure \(\mu\) on K such that its resultant, r(\(\mu)\), equals x, and \(\| \mu \| =p_ K(x)\), where \(p_ K\) denotes the generalized Minkowski functional of K. Such representation is unique, modulo the equivalence relation \(\approx_ K\), \(\forall x\in K\Leftrightarrow K\) is a simplexoid. Theorem 3.2. If K and x are as in theorem 3.1, then there exists a maximal measure \(\mu\) on K such that \(r(\mu)=x\) and \(\| \mu \| =p_ K(x).\) Such representation is unique \(\forall x\in K\Leftrightarrow K\) is a simplexoid. Theorem 3.4. Let K be a closed and bounded absolutely convex subset of a Banach space having the Radon-Nikodým Property, and let \(x\in K\). Then there exists a maximal tight measure \(\mu\) on K such that \(r(\mu)=x\), as a Bochner integral, and \(\| \mu \| =p_ K(x)\). Such representation is unique \(\forall x\in K\Leftrightarrow K\) is a simplexoid. Besides, the author asks two questions about these theorems, and proves some properties, relevant to Choquet theory, of the compact absolutely convex sets and their affine spans.