Definition 5.6.1: Stieltjes Function

Definition 5.6.1 (Stieltjes Function).label Let $F:\R\to\R$, then $F$ is a Stieltjes function if $F$ is right-continuous and non-decreasing. Denote by $\fraks{S}=\fraks{S}(\R,\R)=\curl{F:\R\to\R:F\text{ is Stieltjes}}$ the space of Stieltjes functions and fix $x\in\R$. Define $\fraks{S}_{x}=\curl{F\in\fraks{S}:F(a)=0}$ the space of normalized Stieltjes functions. Let $\sim$ be the equivalence relation on $\fraks{S}$ given by

\begin{align*}\sim:\fraks{S}\ti\fraks{S}\to\fraks{S}&&F\sim G\iff F-G\text{ is constant}\end{align*}

Let $\leq$ be the partial order on $\fraks{S}$ given by

\begin{align*}F\leq G\iff G-F\text{ is non-decreasing}\end{align*}

Then there is an ordered convex cone isomorphism $\fraks{S}_{x}\iso \fraks{S}/\sim$.

Proof. $\frak{S}$ has an action of $(\R,+)$ by translation $c*F=F+c$, for $F\in\fraks{S},x\in\R$. For any $F\in\fraks{S},x\in\R$ we have $\til{F}\define F-F(x)\in\fraks{S}_{x}$ so $\R F\cap\fraks{S}_{x}\neq \emp$. Let $G,H\in \R F\cap\fraks{S}_{x}$ then $G=H+c$ for some $a\in\R$ so

\begin{align*}0=G(x)=H(x)+c=0+c\end{align*}

so $G=H$. By Lemma 5.6.2 $\fraks{S}_{x}\iso \fraks{S}/\sim$ as sets. Since $G-F+c$ is non-decreasing if and only if $G-F$ is non-decreasing for any $c\in\R$ we have by Lemma 5.6.2(1) $\fraks{S}_{x}\iso \fraks{S}/\sim$ as posets.

Let $F,G\in\fraks{S}$, $\al,\be\geq 0$ then

\begin{align*}\al F+\be G\in \fraks{S}\end{align*}

so $\fraks{S}$ is convex cone. Similarly so is $\fraks{S}_{x}$^[1]. By Lemma 5.6.2(2) we have an ordered convex cone isomorphism.$\square$

The quotient inherits a convex cone structure because $\sim$ is quotient by constant functions which is a linear subspacekeyboard_return

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circleLemma 5.6.2

The Eigenspace

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The Eigenspace

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