Let be $S$ a separable(non compact) metric space and $X=C\_b(S)$ the set of all bounded continuous functions, then it's topological dual $X^{\star}=rba(S)$ is the set of all regular Borel additive measures endowed with the variation norm. Denote by $\mathscr{P}(S)$ the subset of $rba(S)$ of all additive probability measures, endowed with the weak$^{\*}$ topology.


Fix $\mu$ a extremal point of $ \mathscr{P}(S)$ and a closed set $\mathcal{F}$ in $\mathscr{P}(S)$ such that the convex hull $co(\mathcal{F})$ do not contains $\mu,$ then it is possible to show that there is an affine functional $\ell\_{\mu,\mathcal{F}}:\mathscr{P}(S)\to \mathbb{R}$ such that $\ell\_{\mu,\mathcal{F}}(\mu)=0$ and $\ell\_{\mu,\mathcal{F}}$ is strictly positive in $\mathcal{F}$


**Question 1:** Is there some way to use the above stated to get a global result, that is, I want to show that there is a affine functional $\ell\_{\mu}:\mathscr{P}(S)\to \mathbb{R}$ such that $\ell\_\mu(\mu)=0$ and $\ell\_\mu(\nu)>0$ for all $\nu\in \mathscr{P}(S)\setminus \{\mu\}$ ?


**Question 2:** If not, is there some other approach to show that there is a affine functional $\ell\_{\mu}:\mathscr{P}(S)\to \mathbb{R}$ such that $\ell\_\mu(\mu)=0$ and $\ell\_\mu(\nu)>0$ for all $\nu\in \mathscr{P}(S)\setminus \{\mu\}$ ?


**Edit:** Following the suggestions of Jochen Glueck I had made some edits in the second paragrapher



Existence of a separating affine functional
Here is a characterization of those $\mu$'s which have the desired property:


**Theorem.** Let $S$ be an arbitrary topoligcal space, let $C\_b(S)$ denote the space of bounded real-valued continuous functios on $S$ (endowed with the supremum norm) and let $\mathscr{P}(S)$ denote the subset of the dual space $C\_b(S)'$ consisting of those functionals which have norm one and map the positive cone $C\_b(S)\_+$ into $[0,\infty)$. For each $\mu \in \mathscr{P}(S)$ the followig assertions are equivalent:


(i) $\mu$ is an *extreme point* of $\mathscr{P}(S)$, i.e. $\mu$ cannot be written as a convex combination of two functionals in $\mathscr{P}(S) \setminus \{\mu\}$.


(ii) $\mu$ is an *exposed point* of $\mathscr{P}(S)$, i.e. there exists a functional $g$ in the bi-dual $C\_b(S)''$ wich attains its strict miminum on the set $\mathscr{P}(S)$ at the point $\mu$.


(iii) There exists a functional $\tilde g$ in the bi-dual $C\_b(S)''$ which is zero at $\mu$ and strictly positive on $\mathscr{P}(S) \setminus \{\mu\}$.


(iv) $\mu$ is a *lattice homomorphism*, i.e. we have $|\langle \mu, f\rangle| = \langle \mu, |f| \rangle$ for all $f \in C\_b(S)$.


(v) $\mu$ is an algebra homomorphism, i.e. we have $\langle \mu, f\_1f\_2 \rangle = \langle \mu, f\_1 \rangle \cdot \langle \mu, f\_2 \rangle$ for all $f\_1,f\_2 \in C\_b(S)$.


*Proof.* "(i) $\Leftrightarrow$ (iv)" This equivalence is a standard argument in the proof of Kakutani's representation theorem for AM-spaces in the theory of Banach lattices; see for instance [H. H. Schaefer: Banach Lattices and Positive Operators (1974), proof of Theorem II.7.4].


"(iii) $\Rightarrow$ (ii)" Obvious.


"(ii) $\Rightarrow$ (iii)" Let $g$ be as in (ii), let $1\_S \in C\_b(S) \subseteq C\_b(S)''$ denote the constant function with value $1$ and set $\tilde g = g - \langle g,\mu\rangle 1\_S$. Then $\tilde g$ fulfils the properties claimed in (iii) (to see this, note that $\langle 1\_S, \nu \rangle = \|\nu\| = 1$ for all $\nu \in \mathscr{P}(S)$).


"(iii) $\Rightarrow$ (i)" Obvious.


"(i) $\Rightarrow$ (iii)" Assume that $\mu$ is an extreme point of $\mathscr{P}(S)$. It follows from Kakutani's representation theorem for AM-spaces [op. cit.] that there exists a Banach lattice isomorphism $\Psi$ between the space $C(K)$ of all real-valued continuous functions on some compact Hausdorff space $K$ and the space $C\_b(S)$ and that this isomorphism can be chosen to fulfil $\Psi(1\_K) = 1\_S$. Now the extreme point property of $\mu$ implies that $\Psi'(\mu) \in C(K)'$ is a delta functional, i.e. there exists a point $\omega \in K$ such that $\Psi'(\mu) = \delta\_\omega$, where $\langle \delta\_\omega,f\rangle = f(\omega)$ for all $f \in C(K)$; this is again a standard fact in Banach lattice theory (see e.g. [P. Meyer-Nieberg: Banach Lattices (1991), Proposition 2.1.2(i)]). The space of all bounded Borel measurable functions on $K$ is contained in the bi-dual of $C(K)$, so the indicator function $1\_{K \setminus \{\omega\}}$ is an element of the bi-dual $C(K)''$ which does not vanish on any positive normalized functional on $C(K)$ except for $\delta\_\omega = \Psi'(\mu)$. Consequently, $\tilde g := \Psi'' 1\_{K \setminus \{\mu\}} \in C\_b(S)''$ does the job in (iii).


"(v) $\Rightarrow$ (iv)" For each $f \in C\_b(S)$ we have
\begin{align\*}
 |\langle \mu, f\rangle|^2 = \langle \mu,f\rangle^2 = \langle \mu, f^2\rangle = \langle \mu, |f|^2 \rangle = \langle \mu,|f|\rangle^2,
\end{align\*}
which implies (iv).


"(iv) $\Rightarrow$ (v)" This can be shown by an approximation procedure which is, for instance, explained in [T. Eisner, B. Farkas, M. Haase, R. Nagel: Operator Theoretic Aspects of Ergodic Theory (2015), supplement to Chapter~7] (though in a slightly different context).


The proof of the theorem is complete.


**Important remark.** The proof of the implication "(i) $\Rightarrow$ (iii)" above should not trick us into believing that every extreme point $\mu$ of $\mathscr{P}(S)$ is of the form $\mu = \delta\_\omega$ for some $\omega \in S$. In fact, the compact space $K$ is usually much larger than $S$ (if, for instance, $S$ is a locally compact space, then the space $K$ is the [Stone–Čech compactification](https://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactification) $\beta S$ of $S$).


