PhyslibAlpha.Mathematics.Convex.Choquet.RepresentingMeasure

Let $S$ be a convex subset of a topological vector space $V$. A probability measure $\mu$ on $S$ is said to represent a point $x \in S$ if, for every continuous linear functional $\ell \colon V \to \mathbb{R}$, the value of $\ell$ at $x$ is equal to the average value of $\ell$ over $S$ with respect to $\mu$, namely: $$\ell(x) = \int_S \ell(y) \, d\mu(y)$$

Let $S$ be a convex subset of a topological vector space $V$. For a point $x \in S$, this is the set of all probability measures $\mu$ on $S$ that represent $x$. A probability measure $\mu$ represents $x$ if, for every continuous linear functional $\ell: V \to \mathbb{R}$, the value of $\ell$ at $x$ is equal to the integral of $\ell$ over $S$ with respect to $\mu$: $$\ell(x) = \int_S \ell(y) \, d\mu(y)$$

Given a topological vector space $V$ over $\mathbb{R}$ and a subset $S \subseteq V$, this function maps a continuous linear functional $\ell: V \to \mathbb{R}$ to its restriction $\ell|_S$, regarded as a continuous function in $C(S, \mathbb{R})$.

Let $S$ be a convex subset of a topological vector space $V$. For any point $x \in S$, the set of probability measures $\mu$ on $S$ that represent $x$ is closed. A probability measure $\mu$ represents $x$ if, for every continuous linear functional $\ell: V \to \mathbb{R}$, the value of $\ell$ at $x$ is equal to the integral of $\ell$ over $S$ with respect to $\mu$: $$\ell(x) = \int_S \ell(y) \, d\mu(y)$$

Let $S$ be a convex set. For two probability measures $\mu$ and $\nu$ on $S$, the Choquet order $\mu \le \nu$ is defined such that for every continuous convex function $f: S \to \mathbb{R}$, the integral of $f$ with respect to $\mu$ is less than or equal to the integral of $f$ with respect to $\nu$: $$\int_S f \, d\mu \le \int_S f \, d\nu$$

Let $S$ be a convex set. For any probability measure $\mu$ on $S$, the Choquet order is reflexive, satisfying $\mu \le \mu$. This implies that for every continuous convex function $f: S \to \mathbb{R}$, the inequality $\int_S f \, d\mu \le \int_S f \, d\mu$ holds.

Let $S$ be a convex set. For any probability measures $\mu, \nu,$ and $\rho$ on $S$, if $\mu \le \nu$ and $\nu \le \rho$ with respect to the Choquet order, then $\mu \le \rho$.

Let $S$ be a convex set and let $\mu$ be a probability measure on $S$. The set of probability measures $\nu$ on $S$ that are greater than or equal to $\mu$ in the Choquet order (denoted $\mu \le \nu$) is a closed set. Here, the Choquet order $\mu \le \nu$ is defined as the condition that for every continuous convex function $f: S \to \mathbb{R}$, the inequality $\int_S f \, d\mu \le \int_S f \, d\nu$ holds.

Let $S$ be a convex subset of a topological vector space. Let $x \in S$ be a point and let $\mu$ and $\nu$ be probability measures on $S$. If $\mu$ represents $x$ (meaning $\ell(x) = \int_S \ell(y) \, d\mu(y)$ for every continuous linear functional $\ell$) and $\mu \le \nu$ in the Choquet order (meaning $\int_S f \, d\mu \le \int_S f \, d\nu$ for every continuous convex function $f$), then $\nu$ also represents $x$.

Let $S$ be a convex subset of a topological vector space. For any point $x \in S$, the set of probability measures $\mu$ on $S$ that represent $x$ is non-empty. A probability measure $\mu$ is said to represent $x$ if for every continuous linear functional $\ell$, the value of $\ell$ at $x$ is the average value of $\ell$ under the measure $\mu$: $$\ell(x) = \int_S \ell(y) \, d\mu(y)$$

Let $S$ be a convex subset of a topological vector space. A probability measure $\mu$ on $S$ is said to be **Choquet maximal at** a point $x \in S$ if $\mu$ represents $x$ and is maximal with respect to the Choquet order among all measures representing $x$. Specifically, this means: 1. For every continuous linear functional $\ell$, the value at $x$ is the integral of $\ell$ with respect to $\mu$: $\ell(x) = \int_S \ell(y) \, d\mu(y)$. 2. For any other probability measure $\nu$ that also represents $x$, if $\mu \le \nu$ in the Choquet order (meaning $\int_S f \, d\mu \le \int_S f \, d\nu$ for all continuous convex functions $f: S \to \mathbb{R}$), then $\nu \le \mu$ in the Choquet order.

Let $S$ be a convex set and $x \in S$ a point. Let $\mathcal{R}(x)$ be the set of probability measures on $S$ that represent $x$ (i.e., those measures whose barycenter is $x$). Let $\mathcal{C} \subseteq \mathcal{R}(x)$ be a chain with respect to the Choquet order $\le$, where $\mu \le \nu$ if $\int_S f \, d\mu \le \int_S f \, d\nu$ for all continuous convex functions $f$ on $S$. For each measure $\mu \in \mathcal{C}$, let $U_\mu = \{ \nu \in \mathcal{R}(x) \mid \mu \le \nu \}$ be the set of representing measures above $\mu$ in the Choquet order. Then the family of sets $\{ U_\mu \}_{\mu \in \mathcal{C}}$ is directed under the superset relation $\supseteq$.

Let $S$ be a convex set and $x \in S$. Let $\mathcal{R}(x)$ be the set of probability measures on $S$ that represent $x$. If $C \subseteq \mathcal{R}(x)$ is a nonempty chain with respect to the Choquet order $\le$, then there exists a probability measure $\mu \in \mathcal{R}(x)$ that is an upper bound for $C$ (i.e., $\nu \le \mu$ for all $\nu \in C$).

Let $S$ be a convex set in a topological vector space. For every point $x \in S$, there exists a probability measure $\mu$ on $S$ that is a Choquet-maximal representing measure for $x$. This means that $\mu$ represents $x$ (the barycenter of $\mu$ is $x$, i.e., $\ell(x) = \int_S \ell \, d\mu$ for all continuous linear functionals $\ell$) and $\mu$ is maximal among all measures representing $x$ with respect to the Choquet order $\preceq$, where $\mu \preceq \nu$ if $\int_S f \, d\mu \le \int_S f \, d\nu$ for every continuous convex function $f: S \to \mathbb{R}$.