Closed convex hull in the space of probability measures In the book Gradient flows: In Metric spaces and in the space of probability measures of Ambrosio, Gigli and Savaré, one can read (Remark 5.1.2) that for $\mathcal K\subset \mathcal P(X)$, one has:
$$\mu\in\overline{\text{Conv}}(\mathcal K)\Leftrightarrow \int_Xfd\mu\leq\sup_{\nu\in\mathcal K}\int_Xfd\nu,\quad \forall f\in C^0_b(X),$$
where $C^0_b(X)$ is the space of bounded continuous functions $f:X\to\mathbf R$.
The authors claim that this comes from Hahn-Banach theorem, but I do not see why...
Can you help me ? For information, I understand the beginning of Remark 5.1.2, namely that narrow convergence is induced by the weak* topology of $(C_b^0(X))'$.
 A: This is a partial solution (in one direction) to the OP. Details about the  notion of weak, weak* and the Hahn-Banach theorems can be found in many books on functional analysis. For example, chapter 3 of Rudin, W. Functional Analysis, 2nd edition, McGraw-Hill, 1973.

Some generalities:
Presumably $\mathcal{P}(X)$ is the space of (Borel) probability measures on a complete separable metric space with the topology inhaireted by the week topology induced by $\mathcal{C}_b(X)$.
$\mathcal{P}(X)$ is a closed and convex subspace of the (locally convex by design) linear space of all complex  measures $\mathcal{M}(X)$ (or only  real signed measures of total finite variation if considering only real linear spaces) with the weak topology induced by $\mathcal{C}_b(X)$. A local basis for this topology are sets of the form
$$\{\mu\in\mathcal{M}(X): |f_k(\mu)|<\varepsilon,\;f_k\in\mathcal{C}_b(X),\,1\leq k\leq n\}$$
for $n\in\mathbb{Z}_+=\mathbb{N}\cup\{0\}$.
Conversely, the dual of $\mathcal{M}(X)$ with the aforementioned weak topology is $\mathcal{C}_b(X)$. Under the  weak* topology on $\mathcal{C}_b(X)$ (induced by $\mathcal{M}(X)$), the dual of $\mathcal{C}_b(X)$ is $\mathcal{M}(X)$.

One direction to the problem in the OP:
By the Hahn-Banach separation theorem (Theorem 3.4(b) in the reference), if $\mu\notin \overline{\operatorname{co}(K)}$ then there is there is a functional $f\in\mathcal{M}^*(X)=\mathcal{C}_b(X)$ and real constants $s<t$ such that
$$\operatorname{Re}(f(\nu))\leq t<s\leq\operatorname{Re}(f(\mu))$$
for all $\nu\in\overline{\operatorname{co}(\mathcal{K})}$.
The real value function $\operatorname{Re}$ may be removed if only real (signed) measures of total finite variations are considered.
