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Cardinality of some convex sets and of their sets of extreme points

Zbigniew Lipecki (2011)

Colloquium Mathematicae

We show that the cardinality of a compact convex set W in a topological linear space X satisfies the condition that ℵ ₀ = . We also establish some relations between the cardinality of W and that of extrW provided X is locally convex. Moreover, we deal with the cardinality of the convex set E(μ) of all quasi-measure extensions of a quasi-measure μ, defined on an algebra of sets, to a larger algebra of sets, and relate it to the cardinality of extrE(μ).

Choquet simplexes whose set of extreme points is K -analytic

Michel Talagrand (1985)

Annales de l'institut Fourier

We construct a Choquet simplex K whose set of extreme points T is 𝒦 -analytic, but is not a 𝒦 -Borel set. The set T has the surprising property of being a K σ δ set in its Stone-Cech compactification. It is hence an example of a K σ δ set that is not absolute.

Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations

Michel Talagrand (1982)

Annales de l'institut Fourier

Let G be a locally compact group. Let L t be the left translation in L ∞ ( G ) , given by L t f ( x ) = f ( t x ) . We characterize (undre a mild set-theoretical hypothesis) the functions f ∈ L ∞ ( G ) such that the map t → L t f from G into L ∞ ( G ) is scalarly measurable (i.e. for φ ∈ L ∞ ( G ) * , t → φ ( L t f ) is measurable). We show that it is the case when t → θ ( L f t ) is measurable for each character θ , and if G is compact, if and only if f is Riemann-measurable. We show that t → L t f is Borel measurable if and only if f is left uniformly continuous.Some of the measure-theoretic tools used there...

Continuous version of the Choquet integral representation theorem

Piotr Puchała (2005)

Studia Mathematica

Let E be a locally convex topological Hausdorff space, K a nonempty compact convex subset of E, μ a regular Borel probability measure on E and γ > 0. We say that the measure μ γ-represents a point x ∈ K if s u p | | f | | ≤ 1 | f ( x ) - ∫ K f d μ | < γ for any f ∈ E*. In this paper a continuous version of the Choquet theorem is proved, namely, if P is a continuous multivalued mapping from a metric space T into the space of nonempty, bounded convex subsets of a Banach space X, then there exists a weak* continuous family ( μ t ) of regular Borel...

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