# Extreme points of subset of set of probability measures

Let $$\mathcal{P}$$ be a set of Borel probability measures on $$[0, 1]$$ equipped with the weak$$^{*}$$-topology as described here: What is the weak*-topology on a set of probability measures?. Consider for $$\rho\in [0, 1]$$, $$\mathcal{P}(\rho)=\left\{\nu\in \mathcal{P}:\int_{0}^{1}{td\nu(t)}=\rho\right\}.$$ I can show that this set is convex and compact for the weak$$^{*}$$-topology. I was wondering what the extreme points of $$\mathcal{P}(m)$$ are and found that the extreme points are of the form $$\nu=p\delta_{x}+q\delta_{y}$$ with $$p, q\geq 0$$, $$x, y\in [0, 1], p+q=1, px+qy=\rho$$ where $$\delta_{x}$$ denotes the Dirac measure at $$x$$. Can someone explain to me why they are of this form?

• Shouldn't $px + py=\rho$?
Apr 27 at 10:45
• You are correct. Thank you! Apr 27 at 10:52

This might not satisfy you, but this is covered by a result of Dubins, cited in my answer to a previous MSE question. You have a compact convex set $$\mathcal P$$ of probability measures, its set of extreme points is the set of point masses. Your $$\mathcal P(\rho)$$ is the intersection of $$\mathcal P$$ and a codimension-1 affine space, namely the measures $$\nu$$ cut out by the single equation $$\int_0^1 td\nu(t)=\rho$$. (This equation is linear in $$\nu$$.) The Dubins result then tells us that the extreme points of $$\mathcal P(\rho)$$ are linear combinations of $$1+1=2$$ extreme points of $$\mathcal P$$, that is, of pairs of point masses.
• Thank your for your answer! Could you elaborate why the condition for $\mathcal{P}$ gives a "linear space of co-dimension 1"? Apr 27 at 11:25