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Consider the set of Borel-measurable probability measures over the interval $[0,1]$ with a given mean, say 1/2. To be precise, I'm talking about the following set $$M=\left(\mu\in \Delta([0,1]):\int x\mu(dx)=1/2\right)$$

This is a convex set, with convex combinations defined as $\mu=\alpha \eta +(1-\alpha)\xi$ when $\mu(E)=\alpha \eta(E)+(1-\alpha)\xi(E)$ for every Borel set $E\subset[0,1]$.

My question is: what is the set of extreme points of this convex set?

My conjecture is that it is the subset of probability measures with support consisting of at most two points. I'm fairly convinced that this is true, but so far I haven't been able to come up with an elegant argument to show this and I thought that perhaps this is a well-known result. It is not hard to show for the case of atomless measures or for the case of discrete measures, but I'd like to have the general case.

flag	asked Jan 12 2012 at 21:24 Henrique de Oliveira 115●4

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It is not too hard to check for the case when the support is of size 3. Then it is not hard to reduce the general case to this case, by partitioning the support into 3 non-null parts. – Ori Gurel-Gurevich Jan 12 2012 at 21:36

Igor Rivin · Accepted Answer · 2012-01-12 22:37:16Z UTC

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This question (where you prescribe a set of moments, on an arbitrary measure space) is completely answered in this very cool paper. (G. Winkler, Extremal points of moment sets).

edited Jan 12 2012 at 22:37

answered Jan 12 2012 at 22:01

Igor Rivin
31.1k●3●37●71

Great reference, thank you so much! – Henrique de Oliveira Jan 12 2012 at 23:05

Extreme points of a set of probability measures

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