Elementary applications of Krein-Milman

functional-analysis convex-analysis locally-convex-spaces

The Krein-Milman theorem is one way to prove De Finnetti's theorem: that every exchangeable sequence of random variables can be seen as a random draw among i.i.d. random variables.

The proof still involves the nontrivial step of showing that the i.i.d. distributions are the extreme points of that set, so it may not be as elementary as you want. It can also be proven in other ways; what's harder is ultimately subjective.

Nevertheless, I think it's a great way to illustrate the power of the theorem, because the statement itself is very easy to understand, and the result is surprising.

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