Keisler measures obtained from hyperfinite samples

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.