I am curious about any property of affine continuous functions from $\mathcal{P}(X) \to \mathbb{R}$ under weak$^*$ topology. Here $X$ is a locally compact Polish space and $\mathcal{P}(X)$ is the set of Borel probability measures on $X$. Affine function $\phi$ satisfies $\phi(\lambda \mu + (1 - \lambda) \nu) = \lambda \phi(\mu) + (1-\lambda) \phi(\nu)$ for any $\mu, \nu \in \mathcal{P}(X)$ and any $\lambda \in [0,1]$.
In particular, it would be super nice for me if any affine continuous functions can be written as linear combination of canonical parings, in other words, if $\phi : \mathcal{P}(X) \to \mathbb{R}$ is affine, I hope $\phi( \mu) = \sum_{i \in I} a_i \int_{X} f_i(x) d\mu(x)$, $a_i \in \mathbb{R}$, $I$ is an index set and $f_i \in \mathcal{C}(X)$, the set of continuous function on $X$.
Is my hope true? If not, any counter example?
