PhyslibAlpha.AlgebraicFramework.OrderUnit.Weight.Basic

Let $E$ be an ordered real vector space. A weight on $E$ is defined as an $\mathbb{R}_{\ge 0}$-linear map from the positive cone $\text{PosCone}(E) = \{x \in E \mid 0 \le x\}$ to the set of extended non-negative real numbers $[0, \infty]$. It assigns a value $w(x) \in [0, \infty]$ to each outcome $x$ in the positive cone such that $w(c_1 x_1 + c_2 x_2) = c_1 w(x_1) + c_2 w(x_2)$ for all $c_1, c_2 \in \mathbb{R}_{\ge 0}$ and $x_1, x_2 \in \text{PosCone}(E)$.

Let $E$ be an ordered real vector space. For any two weights $w_1$ and $w_2$ on $E$, if $w_1(x) = w_2(x)$ for every element $x$ in the positive cone $\text{PosCone}(E)$, then $w_1 = w_2$.

Let $E$ be an ordered real vector space and let $w$ be a weight on $E$, which is an $\mathbb{R}_{\ge 0}$-linear map from the positive cone $\text{PosCone}(E)$ to the extended non-negative real numbers $[0, \infty]$. Then $w(0) = 0$, where $0$ denotes the zero element of $\text{PosCone}(E)$.

Let $E$ be an ordered real vector space and $\text{PosCone}(E) = \{z \in E \mid 0 \le z\}$ be its positive cone. For any weight $w$ on $\text{PosCone}(E)$ and any elements $x, y \in \text{PosCone}(E)$, the weight of their sum is the sum of their individual weights, i.e., $w(x + y) = w(x) + w(y)$.

Let $E$ be an ordered real vector space and $\text{PosCone}(E) = \{x \in E \mid 0 \le x\}$ be its positive cone. For any weight $w: \text{PosCone}(E) \to [0, \infty]$, the map is monotone: for any $x, y \in \text{PosCone}(E)$, if $x \le y$, then $w(x) \le w(y)$. In other words, a larger outcome in the positive cone never receives a smaller weight.

A weight $w$ on the positive cone $\text{PosCone}(E)$ is **faithful** if for every element $x \in \text{PosCone}(E)$, $w(x) = 0$ implies $x = 0$. Intuitively, this means that only the impossible outcome (the zero element) carries no weight at all.

Let $E$ be an ordered real vector space and $\text{PosCone}(E) = \{x \in E \mid x \ge 0\}$ be its positive cone. A weight $w : \text{PosCone}(E) \to [0, \infty]$ is said to be finite if for every outcome $x \in \text{PosCone}(E)$, the value assigned by the weight is not infinite, i.e., $w(x) < \infty$.

A weight $w$ on the positive cone $\text{PosCone}(E)$ of an ordered real vector space $E$ is **semifinite** if for every non-zero outcome $x \in \text{PosCone}(E)$, there exists a non-zero outcome $y \in \text{PosCone}(E)$ such that $y \le x$ and its weight $w(y)$ is finite ($w(y) < \infty$). This property ensures that every non-zero outcome, however large, contains some piece of finite weight.

A weight $w$ is said to be normal if it preserves the suprema of directed sets. Specifically, for any non-empty directed set $D \subseteq \text{PosCone}(E)$ that has a least upper bound $x = \sup D$, the weight of the supremum is the supremum of the weights: $w(x) = \sup \{w(d) \mid d \in D\}$ in the extended non-negative real numbers $[0, \infty]$.

Given two weights $w_1, w_2$ on the positive cone $\text{PosCone}(E)$ and two non-negative real scalars $a, b \in \mathbb{R}_{\ge 0}$, their mixture is the weight $a w_1 + b w_2$ defined by $(a w_1 + b w_2)(x) = a w_1(x) + b w_2(x)$ for every $x \in \text{PosCone}(E)$. This operation results in a new weight that maps outcomes to values in the extended non-negative real numbers $[0, \infty]$.

Let $E$ be an ordered real vector space and $\text{PosCone}(E)$ be its positive cone. For any weights $w_1, w_2$ on $\text{PosCone}(E)$, non-negative real scalars $a, b \in \mathbb{R}_{\ge 0}$, and any outcome $x \in \text{PosCone}(E)$, the value of the mixture of $w_1$ and $w_2$ with coefficients $a$ and $b$ applied to $x$ is equal to $a w_1(x) + b w_2(x)$.

The certain outcome is represented by the identity element $1$ of the space $E$, considered as an element of the positive cone $\text{PosCone}(E)$.

Let $E$ be an ordered real vector space with an order unit $\mathbb{1} \in \text{PosCone}(E)$. Suppose $w_1$ and $w_2$ are weights on the positive cone of $E$ that are states (meaning they are finite and normalized such that $w_1(\mathbb{1}) = 1$ and $w_2(\mathbb{1}) = 1$). For any non-negative real coefficients $a, b \in \mathbb{R}_{\ge 0}$ such that $a + b = 1$, the mixture $a w_1 + b w_2$ is also a state.

Let $E$ be an ordered real vector space and $\text{PosCone}(E) = \{x \in E \mid x \ge 0\}$ be its positive cone. A weight $w : \text{PosCone}(E) \to [0, \infty]$ is finite (that is, $w(x) < \infty$ for all $x \in \text{PosCone}(E)$) if and only if its value at the order unit $1 \in \text{PosCone}(E)$ is finite, i.e., $w(1) < \infty$. This equivalence holds because the forward implication is immediate, and conversely, every element in the positive cone is bounded by a natural multiple of the order unit, allowing the finiteness to transfer via monotonicity.

For a weight $w$ on the positive cone $\text{PosCone}(E)$, its normalization is a weight defined by scaling $w$ by the multiplicative inverse of its value at the order unit $1 \in \text{PosCone}(E)$. Specifically, for any element $x \in \text{PosCone}(E)$, the normalized weight assigns the value $(w(1))^{-1} \cdot w(x)$, where $w(1)$ is considered the "mass" of the weight. This construction produces a normalized state when the mass $w(1)$ is finite and non-zero.

Let $E$ be an ordered real vector space. For any weight $w$ on the positive cone $\text{PosCone}(E)$ and any element $x \in \text{PosCone}(E)$, the value of the normalized weight $\text{normalize}(w)$ at $x$ is given by the formula $$(\text{normalize } w)(x) = (w(1))^{-1} \cdot w(x),$$ where $1$ denotes the order unit in $\text{PosCone}(E)$.

Let $E$ be an ordered real vector space and $w$ be a weight on the positive cone $\text{PosCone}(E)$. If $w$ is finite (i.e., $w(x) < \infty$ for all $x \in \text{PosCone}(E)$) and its mass at the order unit $1 \in \text{PosCone}(E)$ is non-zero ($w(1) \neq 0$), then the normalized weight $\text{normalize}(w)$ is a state weight.