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True: life is nice in the metrizable case. A very good reference is Phelp's book Lectures on the Choquet's theorem (LNM). On the contrary, note that for

For a non-metrizable compact convex subset of a locally convex space, extreme points need not even form a Borel set. This has been shown by Bishop-de Leeuw, The representation of linear functionals by measures on sets of extreme points, Ann.Inst. Fourier (Grenoble) (1959) . A very good reference for these topics is Phelp's LNM Lectures on the Choquet's theorem (2001).
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True: life is nice in the metrizable case. A very good reference is Phelp's book Lectures on the Choquet's theorem (LNM). Note On the contrary, note that for a nonmetrizable non-metrizable compact convex subset of a locally convex space, extreme points need not even form a Borel set. This has been shown by Bishop-de Leeuw, The representation of linear functionals by measures on sets of extreme points, Ann.Inst. Fourier (Grenoble) (1959) .
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A very good reference is Phelp's book Lectures on the Choquet's theorem (LNM). Note that for a nonmetrizable compact convex subset of a locally convex space, extreme points need not even form a Borel set. This has been shown by Bishop-de Leeuw, The representation of linear functionals by measures on sets of extreme points, Ann.Inst. Fourier (Grenoble) (1959) .