PhyslibAlpha.Mathematics.Convex.Choquet.BoundaryRepresentation

Given a family of real-valued functions (tests) $f_a: X \to \mathbb{R}$ indexed by $a \in A$, a map $\iota: B \to X$ from a boundary space $B$, and a point $x \in X$, a measure $\mu$ on $B$ is a boundary representation of $x$ if $\mu$ is a regular probability measure such that for every $a \in A$, the integral of the test function over the boundary agrees with its value at $x$: $$\int_{B} f_a(\iota(b)) \, d\mu(b) = f_a(x)$$

Given a family of real-valued test functions $f_a: X \to \mathbb{R}$ indexed by $a \in A$ and a map $\iota: B \to X$ from a boundary space $B$, a point $x \in X$ has a **boundary decomposition** if there exists a regular probability measure $\mu$ on $B$ that represents $x$. This means that for every $a \in A$, the integral of the test function over the boundary agrees with its value at $x$: $$\int_{B} f_a(\iota(b)) \, d\mu(b) = f_a(x)$$

Given a family of real-valued functions (tests) $f_a: X \to \mathbb{R}$ indexed by $a \in A$, a map $\iota: B \to X$ from a boundary space $B$, and a point $x \in X$, the point $x$ has a unique boundary decomposition if there exists exactly one regular probability measure $\mu$ on $B$ such that for every $a \in A$, the average of the test function over the boundary agrees with its value at $x$: $$\int_{B} f_a(\iota(b)) \, d\mu(b) = f_a(x)$$

A family of real-valued test functions $f_a: X \to \mathbb{R}$ indexed by $a \in A$ and a map $\iota: B \to X$ from a boundary space $B$ form a **simplex representation** if every point $x \in X$ has a unique boundary decomposition. That is, for every $x \in X$, there exists exactly one regular probability measure $\mu$ on $B$ such that for all $a \in A$, the average of the test function over the boundary agrees with its value at $x$: $$\int_{B} f_a(\iota(b)) \, d\mu(b) = f_a(x)$$