PhyslibAlpha.AlgebraicFramework.Measurement.BoundedScalarization

This instance equips the space $E$ with the structure of a normed additive commutative group. The norm $\|x\|$ is defined as the order-unit norm induced by an Archimedean order unit $\mathbb{1}$, calculated as $\|x\| = \inf \{r \in \mathbb{R}_{\ge 0} \mid -r \cdot \mathbb{1} \le x \le r \cdot \mathbb{1} \}$.

This definition equips the space $F$ with the structure of a normed additive commutative group. The norm $\|x\|$ for an element $x \in F$ is the order-unit norm, which is defined as the infimum of all $r \in \mathbb{R}_{\ge 0}$ such that $-r \cdot 1 \le x \le r \cdot 1$, where $1$ is the Archimedean order unit of $F$. This structure induces a metric on $F$ where the distance between two elements $x$ and $y$ is given by $\|x - y\|$.

Let $\Omega$ be a measurable space, and let $E$ and $F$ be ordered vector spaces equipped with order-unit norms. Let $f: \Omega \to \mathbb{R}$ be a measurable function that is bounded by some constant $M \in \mathbb{R}$ (i.e., $|f(x)| \leq M$ for all $x \in \Omega$). Let $\mu$ be an effect-valued measure on $\Omega$ taking values in $E$, and let $\phi: E \to F$ be a normal unital positive linear map. Then the integral of $f$ commutes with the action of the map $\phi$: $$\phi \left( \int_\Omega f \, d\mu \right) = \int_\Omega f \, d(\phi \circ \mu)$$ where $\phi \circ \mu$ is the push-forward effect-valued measure defined by $(\phi \circ \mu)(S) = \phi(\mu(S))$ for any measurable set $S \subseteq \Omega$.