PhyslibAlpha.AlgebraicFramework.JordanOrderUnit.ProjectionResolution

Any measurable projection resolution $P$ defined on a measurable space $\Omega$ with values in an effect space $E$ can be canonically viewed as an effect-valued measure.

This instance allows a measurable projection resolution $P$ to be treated as a function. For any set $s \subseteq \Omega$ and a proof $h$ that $s$ is a measurable set, the function returns an effect $P(s, h)$ in $E$, which is an element $e \in E$ such that $0 \le e \le 1$.

Let $\Omega$ be a measurable space and $E$ be an additive commutative group equipped with a partial order and a unit element $1$. Let $P$ and $Q$ be measurable projection resolutions on $\Omega$ taking values in the set of effects $[0, 1] \subseteq E$. If for every measurable set $s \subseteq \Omega$, the values assigned by the resolutions are equal, $P(s) = Q(s)$, then the resolutions themselves are equal, $P = Q$.

Let $P$ be a measurable projection resolution on a measurable space $\Omega$ with values in the set of effects $E \subseteq [0, 1]$. For the empty set $\emptyset$, the resolution assigns the zero effect $0$, i.e., $P(\emptyset) = 0$.

Let $\Omega$ be a measurable space and $E$ be an ordered additive commutative group with a unit element $1$. For any measurable projection resolution $P$ from $\Omega$ to the set of effects in $E$, the value of the resolution at the universal set $\Omega$ is the unit effect $1$: $$P(\Omega) = 1.$$

Let $\Omega$ be a measurable space and $E$ be a Jordan algebra. For any measurable projection resolution $P$ on $\Omega$ with values in $E$ and any measurable set $s \subseteq \Omega$, the element $P(s)$ is a Jordan projection. That is, $P(s)$ is idempotent with respect to the Jordan product: $$P(s) \circ P(s) = P(s).$$

Let $\Omega$ be a measurable space and $E$ be a Jordan algebra (equipped with an additive commutative group structure, a partial order, and a unit element $1$). Let $P$ be a measurable projection resolution from $\Omega$ to $E$, which maps measurable sets to effects in $E$. For any two measurable sets $s, t \subseteq \Omega$, the Jordan product of their corresponding projections $P(s)$ and $P(t)$ is equal to the projection of their intersection $s \cap t$: $$P(s) \cdot P(t) = P(s \cap t).$$

Let $P$ be a measurable projection resolution from a measurable space $\Omega$ to a Jordan algebra $E$. If $s$ and $t$ are disjoint measurable sets in $\Omega$, then their corresponding projections $P(s)$ and $P(t)$ in $E$ are Jordan-orthogonal: $$P(s) \circ P(t) = 0.$$

Let $(\Omega, \Sigma)$ be a measurable space and $E$ be an ordered Jordan algebra (an additive commutative group with a partial order, a unit element $1$, and a Jordan product). Let $P$ be a measurable projection resolution from $\Omega$ to $E$, which assigns to each measurable set $s \in \Sigma$ an effect $P(s) \in [0, 1] \subseteq E$. For any sequence of pairwise disjoint measurable sets $\{s_n\}_{n \in \mathbb{N}} \subseteq \Sigma$, the projection of their countable union is the least upper bound (supremum) of the sequence of partial sums of their individual projections. That is, $$P\left(\bigcup_{n=0}^\infty s_n\right) = \sup \left\{ \sum_{n=0}^{N-1} P(s_n) : N \in \mathbb{N} \right\}$$ holds in the partial order of $E$.

This definition establishes a normed additive commutative group structure on the space $E$ associated with a measurable projection resolution. The topology is induced by the order-unit norm $\|\cdot\|$, which is defined for any element $x \in E$ as the infimum of all $r \in \mathbb{R}_{\ge 0}$ such that $-r \cdot \mathbf{1} \le x \le r \cdot \mathbf{1}$, where $\mathbf{1}$ is the Archimedean order unit of $E$.

Let $P$ be a measurable projection resolution on a measurable space $\Omega$ with values in an ordered Jordan algebra $E$. Given a measurable function $f: \Omega \to \mathbb{R}$ and a bound $M \in \mathbb{R}$ such that $|f(x)| \leq M$ for all $x \in \Omega$, the **bounded Borel calculus** of $f$ with respect to $P$ is defined as the integral of $f$ against the underlying effect-valued measure of $P$, denoted as $\int_\Omega f \, dP$. This value is an element of $E$ obtained as the limit of the integrals of simple functions that uniformly approximate $f$.

Let $P$ be a measurable projection resolution on a measurable space $\Omega$ with values in an ordered Jordan algebra $E$. If $f: \Omega \to \mathbb{R}$ is a measurable function that is bounded (there exists $M \in \mathbb{R}$ such that $|f(x)| \le M$ for all $x \in \Omega$) and pointwise non-negative ($f(x) \ge 0$ for all $x \in \Omega$), then the bounded Borel calculus $\int_\Omega f \, dP$ is a non-negative element of $E$, i.e., $0 \le \int_\Omega f \, dP$.

Let $P$ be a measurable projection resolution on a measurable space $\Omega$ with values in an ordered Jordan algebra $E$. For any measurable function $f: \Omega \to \mathbb{R}$ and any two real constants $M, M' \in \mathbb{R}$ such that $|f(x)| \leq M$ and $|f(x)| \leq M'$ for all $x \in \Omega$, the bounded Borel calculus $\int_\Omega f \, dP$ is independent of whether $M$ or $M'$ is used as the uniform bound in its construction.

Let $P$ be a measurable projection resolution on a measurable space $\Omega$ with values in an ordered Jordan algebra $E$. Let $f, g: \Omega \to \mathbb{R}$ be measurable functions such that $|f(x)| \leq M$ and $|g(x)| \leq M'$ for some constants $M, M' \in \mathbb{R}$ and all $x \in \Omega$. Then the bounded Borel calculus satisfies the additivity property: $$\int_\Omega (f + g) \, dP = \int_\Omega f \, dP + \int_\Omega g \, dP$$ where $\int_\Omega f \, dP$ denotes the bounded Borel calculus of $f$ with respect to $P$.

Let $P$ be a measurable projection resolution on a measurable space $\Omega$ with values in an ordered Jordan algebra $E$. For any scalar $c \in \mathbb{R}$ and measurable function $f: \Omega \to \mathbb{R}$ such that $|f(x)| \le M$ for all $x \in \Omega$, the bounded Borel calculus (denoted as the integral $\int_\Omega f \, dP$) is real-homogeneous. That is, $$\int_\Omega (c \cdot f) \, dP = c \cdot \int_\Omega f \, dP,$$ where $c \cdot f$ is the function $x \mapsto c \cdot f(x)$, and the bound for $c \cdot f$ is taken as $|c| \cdot M$.

Let $P$ be a measurable projection resolution on a measurable space $\Omega$ taking values in an ordered Jordan algebra $E$. For any measurable set $s \subseteq \Omega$, the bounded Borel calculus of the indicator function $\mathbb{1}_s$ with respect to $P$ is equal to the projection $P(s)$ in $E$. That is, $$\int_\Omega \mathbb{1}_s \, dP = P(s).$$

Let $P$ be a measurable projection resolution on a measurable space $\Omega$ with values in an ordered Jordan algebra $E$. The bounded Borel calculus of the constant function $f(\omega) = 1$ for all $\omega \in \Omega$ is equal to the order unit $\mathbf{1}$ of $E$, denoted as: $$\int_\Omega 1 \, dP = \mathbf{1}$$ This expresses the normalization property of the underlying effect-valued measure.

Let $P$ be a measurable projection resolution on a measurable space $\Omega$ taking values in an ordered Jordan algebra $E$. For a measurable function $f: \Omega \to \mathbb{R}$ that is bounded such that $0 \le f(x) \le 1$ for all $x \in \Omega$, the bounded Borel integral $\int_\Omega f \, dP$ is an **effect** in $E$, meaning it satisfies $0 \le \int_\Omega f \, dP \le 1$.

Let $P$ be a measurable projection resolution on a measurable space $\Omega$ with values in an ordered Jordan algebra $E$. For any measurable function $f : \Omega \to \mathbb{R}$ such that $0 \le f(x) \le 1$ for all $x \in \Omega$, the bounded Borel integral $\int_\Omega f \, dP$ constructed as an effect (an element in $[0, 1] \subseteq E$) is equal to the integral of $f$ against $P$ obtained via the general bounded Borel calculus with bound $M=1$.

Let $P$ and $Q$ be two measurable projection resolutions on a measurable space $\Omega$ with values in an ordered Jordan algebra $E$. If for every measurable function $f : \Omega \to \mathbb{R}$ that is bounded (i.e., there exists a constant $M \in \mathbb{R}$ such that $|f(x)| \le M$ for all $x \in \Omega$), the bounded Borel calculus integrals are equal, $\int_\Omega f \, dP = \int_\Omega f \, dQ$, then $P = Q$. This theorem establishes that the bounded Borel calculus uniquely determines a projection resolution.