I am looking for an example:

Suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is a probability space, where $\Omega\subseteq \mathbb{R}^n$ (for some $n\in \mathbb{N}$) is a totally ordered set with respect to the relation $\le_R$ (which is not necessarily the lexicographical order). We assume the sigma algebra $\mathcal{F}$ contains all sets of the form $\{z\in \Omega : z <_R x\}$ and $\{z\in \Omega : z\le_R x\}$ for $x\in \Omega$. Is there a set $S\in \mathcal{F}$ with positive measure such that $\sup_{x, y\in S} \mathbb{P}\{z\in S: x\le_R z\le_R y\} = 0$.

If $(S, \le_R)$ has both countable coinitiality and countable cofinality, then this situation will not happen. Is there such an example in the general cases?