- #1

Fredrik

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## Main Question or Discussion Point

Let [itex]\mathcal B(\Omega)[/itex] be the Borel algebra of [itex]\Omega[/itex] (the σ-algebra of Borel sets in [itex]\Omega[/itex]). I understand that if we define a "convex combination" of probability measures by [tex]\bigg(\sum_{k=1}^n w_k\mu_k\bigg)(E)=\sum_{k=1}^n w_k\mu_k(E),[/tex] then every convex combination of probability measures is a probability measure. I'm particularly interested in the probability measures defined in the following way: For each [itex]s\in\Omega[/itex], we define [tex]\mu_s(E)=\chi_E(s)=\begin{cases}1 & \text{ if }s\in E\\ 0 & \text{ if }s\notin E\end{cases}[/tex]

Let [itex]S_0[/itex] be the set of all probability measures defined this way, and let [itex]S[/itex] be the set of all convex combinations of members of [itex]S_0[/itex]. I have found that S is closed under convex combinations.

My question is: Are there any probability measures on [itex]\mathcal B(\Omega)[/itex] that aren't members of [itex]S[/itex]?

Let [itex]S_0[/itex] be the set of all probability measures defined this way, and let [itex]S[/itex] be the set of all convex combinations of members of [itex]S_0[/itex]. I have found that S is closed under convex combinations.

My question is: Are there any probability measures on [itex]\mathcal B(\Omega)[/itex] that aren't members of [itex]S[/itex]?