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On affine extensions of continuous functions defined on the extreme boundary of a Choquet simplex. (English) Zbl 0582.46010
The author shows that if the set of extreme points \(\partial_ eK\) of a Choquet simplex K is Lindelöf, then every bounded continuous real- valued function on \(\partial_ eK\) may be extended to a bounded real- valued affine function on K of the first Baire class. This result subsumes the known results concerning the metrizable and K-analytic cases. The proof is based upon an interesting lemma:
Let X be a Lindelöf space, f a continuous function on X and F be a downward directed family of continuous functions on X such that \(f(x)=\inf \{h(x):\) \(h\geq f\}\) for each x in X. Then there is a decreasing sequence \((g_ n)\) in F such that \(g_ n\) converges pointwise to f.
Reviewer: S.A.Naimpally

MSC:
46A55 Convex sets in topological linear spaces; Choquet theory
54C50 Topology of special sets defined by functions
54C30 Real-valued functions in general topology
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