Existence of a separating affine functional 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
 A: 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 $\beta S$ of $S$).
A: In fact, extremal points of $\mathcal{P}(S)$ are exactly the deltas (if $\operatorname{supp}(\mu)$ is not a single point $p$, it can easily be written as a mean of two probability measures). In this case, any  nonnegative $f\in C_b(S)$ vanishing exactly at $p $  easily does the job, because then $\int_S fd\nu>0$ for all $\nu\in \mathcal{P}(S)\setminus\{\mu\}.$
