Cyclically Monotone Subsets of the Doubled Real Line: Two-Point Monotonicity, Ordering of the Sections, and Countability of the Multi-Valued Abscissae

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, in dimension $d=1$. As in Borel Sigma-Algebra on Euclidean Space and the preamble of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets, the real line $\mathbb{R}$ is identified with the Euclidean space $\mathbb{R}^{1}$, a point of $\mathbb{R}^{1}$ being read as its sole coordinate, so that $\mathcal{B}(\mathbb{R}^{1})=\mathcal{B}(\mathbb{R})$ and $\mathbb{R}^{1+1}=\mathbb{R}^{2}$; by claim 1 of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative one has $x\cdot y=xy$ for $x,y\in\mathbb{R}^{1}$. Write $\mathrm{pr}_{1}=\mathrm{pr}^{1,1}_{1}$ and $\mathrm{pr}_{2}=\mathrm{pr}^{1,1}_{2}$ for the coordinate projections $\mathbb{R}^{2}\to\mathbb{R}$.

Let $\Gamma\subseteq\mathbb{R}^{2}$ be cyclically monotone. For $x\in\mathbb{R}$ the section of $\Gamma$ over $x$ is the set

$$\Gamma_{x}=\{\mathrm{pr}_{2}(z)\ :\ z\in\Gamma\text{ and }\mathrm{pr}_{1}(z)=x\}\subseteq\mathbb{R},$$

and $M_{\Gamma}$ denotes the set of those $x\in\mathbb{R}$ for which $\Gamma_{x}$ has two distinct elements.

1. (Two-point monotonicity)(i) For all $z,z'\in\Gamma$,

$$0\le\bigl(\mathrm{pr}_{2}(z)-\mathrm{pr}_{2}(z')\bigr)\bigl(\mathrm{pr}_{1}(z)-\mathrm{pr}_{1}(z')\bigr).$$

2. (Ordering of the sections)(ii) Let $x,x'\in\mathbb{R}$ with $x<x'$, let $y\in\Gamma_{x}$ and let $y'\in\Gamma_{x'}$. Then $y\le y'$.

3. (Countably many multi-valued abscissae)(iii) The set $M_{\Gamma}$ is countable, and for every $x\in\mathbb{R}\setminus M_{\Gamma}$ the section $\Gamma_{x}$ is empty or has exactly one element.