# “Inner Regularity” of probability measure on totally ordered sets

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?

• Is $\le$ the usual ordering of the reals, or did you mean $\le_R$? – Andrés E. Caicedo Oct 7 at 2:16
• In case you meant $\le_R$: Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1$ to the cocountable ones. (Cont.) – Andrés E. Caicedo Oct 7 at 2:18
• (cont.) For any $x,y\in\Omega$ with $x\le_R y$, the interval $\{z\in \Omega: x\le_R z\le_R y\}$ has measure 0, so you can take as $S$ any cocountable subset of $\Omega$. – Andrés E. Caicedo Oct 7 at 2:18
• I meant $\le_R$. Thank you for your post, I will think about it. – Ray Oct 7 at 2:30
• Hmm, how is it relevant to your question that $\Omega$ is a subset of $\mathbb{R}^n$? None of the properties you discuss seem to be related to any properties of the spaces $\mathbb{R}^n$ (except that it bounds the cardinality of $\Omega$). – Jochen Glueck Oct 7 at 6:07

## 1 Answer

Consider a subset $$\Omega$$ of $$\mathbb R$$ of size $$\aleph_1$$ and ordered in type $$\omega_1$$. (This uses the axiom of choice.)

Let $$\mathcal F$$ be the $$\sigma$$-algebra generated by the initial segments of $$\Omega$$ under the well-ordering (so all sets in $$\mathcal F$$ are countable or co-countable), with the measure that assigns $$0$$ to the countable sets and $$1$$ to the cocountable ones.

For any $$x,y\in\Omega$$ with $$x\le_R y$$, the interval $$\{z\in \Omega: x\le_R z\le_R y\}$$ has measure 0, so you can take as $$S$$ any cocountable subset of $$\Omega$$.