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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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