Total convex structures and ultraproducts

Let $(\mathscr{E} ,S)$ be an event structure [1]. A set of states $S$ is said to be a convex structure if it has the following two properties:
(1) for any positive numbers $\lambda_1,\lambda_2,\dots,\lambda_n$ satisfying the condition $\sum^n_{i =1}\lambda_i = 1$ and any states $s_1, s_2,\dots, s_n$ there is a unique element $\langle\lambda_1,\lambda_2,\dots,\lambda_n; s_1, s_2,\dots, s_n\rangle\in S$; (2) $\langle\lambda_1,\lambda_2,\dots,\lambda_n; s, s,\dots, s\rangle = s$.
This state $\langle\lambda_1,\lambda_2,\dots,\lambda_n; s_1, s_2,\dots, s_n\rangle$ is called a mixture of states $s_1, s_2,\dots, s_n$. For mixtures of two states, we use the notation $\langle\lambda,1-\lambda; s, t\rangle = \langle\lambda; s, t \rangle$. A state $s \in S$ is said to be pure if it cannot be written as $s = \langle\lambda; t_1, t_2\rangle$ for some $t_1 \neq t_2$.
We define the distance function of close states $\sigma(s, t)$ as follows: if there exist $s_1, t_1 \in S$ such that the condition $\langle\lambda; s_1, s\rangle= \langle\lambda; t_1, t \rangle$ holds, then $\sigma(s, t ) = \inf \{0 < \lambda \le 1 : \langle\lambda; s_1, s\rangle = \langle\lambda; t_1, t \rangle\}$ ; otherwise, $\sigma(s, t ) = 1/2$. In general, this function is not a metric.
A convex structure $S$ is said to be a $\sigma$-convex structure if the following two properties hold: (1) If $s_n \in S$ and $\lim_{n,m\to\infty}\sigma(s_n, s_m) = 0$, then there exists a unique $s \in S$ such that $\lim_{n\to\infty}\sigma(s_n, s) = 0$; (2) If $\lambda_i > 0, \sum\lambda_i = 1, t_1, t_2,\dots \in S$ and $s_n = \langle\lambda_1,\dots,\lambda_n,1-\sum^{\infty}_{i=n+1}; t_1,\dots, t_{n+1}\rangle$, then $\lim_{n,m\to\infty}\sigma(s_n, s_m) = 0$.
Thus, we can consider infinite (countable) mixtures of states.
A map $f : S \to \mathbb{R}$ is said to be an affine functional if we have $f (\langle\lambda_1,\lambda_2,\dots,\lambda_n; s_1, s_2,\dots, s_n\rangle) = \sum^n_{i=1}\lambda_i s_i$ for any sets $\lambda_1,\lambda_2,\dots,\lambda_n; s_1, s_2,\dots, s_n, \lambda_i>0, \sum\lambda_i = 1$. Note that the set of affine functionals $S^*$ is a linear space with respect to pointwise operations. We define «zero» and «identity» affine functionals: $\mathbf{0}(s) = 0, \mathbf{1}(s) = 1$ for all $s \in S$. Also we define a partial order relation on $S^*$: $f \le g \Leftrightarrow f(s) \le g(s)$ for all $s \in S$. The functional $f \in S^*$ is called an effect if $\mathbf{0} \le f \le \mathbf{1}$. The set of effects will be denoted by $E(S)$. It forms a convex subset of the linear space $S^*$. A total convex structure is a $\sigma$-convex structure with the property: $f (s) = f (t )$ for every effect $f$ implies that $s = t$ . It is well known [1] that then $(S,\sigma)$ is a complete metric space.
Let $(\mathscr{E}_n,S_n)_{n\ge 1}$ be a sequence of event structures, $\mathscr{U}$ be a non-trivial ultrafilter in the set $\mathbb{N}$ of naturals. Then the ultraproduct $((\mathscr{E}_n)_{\mathscr{U}} ,S_{\mathscr{U}} )$ is an event structure [2]. On the set of states $S_{\mathscr{U}}$ we introduce by means of factors a total convex structure and a set of effects.
Theorem 1. Let $(S_n,\sigma_n)_{n\ge 1}$ be a sequence of total convex structures, $\mathscr{U}$ be a nontrivial ultrafilter in set of natural numbers $\mathbb{N}$. Then the ultraproduct $(S_{\mathscr{U}} ,\sigma_{\mathscr{U}} )$ is a total convex structure.
Next, we consider the properties of ultraproducts of observables on the set of effects $E(S)$.