Keisler measures obtained from hyperfinite samples

Question

Let $M$ be an infinite model.

Let $F$ be the set of maps $M\to\mathbb R^+\cup\{0\}$ with finite support (i.e. 0 almost everywhere).

Let $\langle M,\mathbb R, F\rangle$ a 3-sorted expansion of $M$ in a language that contains (most relevantly) a function symbol $\Sigma_{\varphi(x,y)}$ for every formula $\varphi(x,y)$ with the following interpretation

$\Sigma_{\varphi(x,y)}\ :\ F\times M^{|y|}\to\mathbb R$

$\qquad\qquad (f,b)\ \ \ \ \mapsto\displaystyle\sum_{\varphi(x,b)} fx.$

Let $\langle\mathcal U, {}^*\mathbb R, \mathcal F\rangle$ be a saturated extension of $\langle M,\mathbb R, F\rangle$.

Given $f\in\mathcal F$, there is an obvious way to construct a global Keisler measure $\mu_f$.

Question: is there a natural characterization of the measures $\mu_f$ of this form?

I am interested in the the general case and when Th$(M)$ is NIP.

Other definitions and minor details

1. The language of $\langle M,\mathbb R, F\rangle$ has also all possible relations and functions of $\mathbb R$ and of $F$.

2. A global Keisler measure is a finitely additive probability measure on the formulas $\varphi(x)$ with parameters in $\mathcal U$. For simplicity, $|x|=1$.

3. Given any nonnull $f\in\mathcal F$ we define $\mu_f\big(\varphi(x,b)\big)$ as the standard part of

$$\sum_{\varphi(x,b)} fx \bigg/ \sum_{x=x} fx$$

Answer 1

I accidentally stumbled on this old question of mine. I add my own answer (or, should I better erase the question?).

every Keiser measure is of the form $\mu_f$ for some $f$. (You even need not take the standard part.)

Let $u$ be a variables of sort $\mathcal F$. We claim that the type $p(u)$ below is finitely consistent

$$p(u)=\Big\{\sum_{\varphi(x)}ux=\mu\varphi(x)\quad:\ \ \varphi(x)\in\Delta\Big\}$$

Let $\{\varphi_1(x),\dots,\varphi_n(x)\}\subseteq\Delta$. It suffices to show that there is $f\in\mathcal F$ such that for all $i=1,\dots,n$.

$$\tag{1}\sum_{\varphi_i(x)}fx=\mu\varphi_i(x)$$

Without loss of generality we can assume that $\{\varphi_1(x),\dots,\varphi_n(x)\}$ is a Boolean algebra with atoms $\varphi_1(x),\dots,\varphi_k(x)$ for some $k\le n$. Pick some $a_1,\dots,a_k\in\mathcal U$ such that $a_i\models\varphi_i(x)$. Pick $f\in\mathcal F$ with support $\{a_1,\dots,a_k\}$ such that for all $i=1,\dots,k$

$$f(a_i)=\mu\varphi_i(x)$$ .

By the finite additivity of $\mu$, equation (1) above is satisfied.