Let $(X,\mathcal{A})$ be a measurable space, and let $\{\le\}$ be a partial order relation on $X$ such that, seen as a subset of $X\times X$, is measurable for the product $\sigma$-algebra. (This has a number of consequences, for example that the diagonal is measurable.)
Consider now the down-sets $$ \downarrow y = \{x\in X : x\le y\} . $$
When can we say that these sets generate the $\sigma$-algebra $\mathcal{A}$?
- An example where this happens is the real line with its Borel $\sigma$-algebra and usual order.
- An example where this does not happen is the real line with its Borel $\sigma$-algebra and the discrete order (= diagonal in $X\times X$). [Thanks to Dabouliplop for pointing this out.]
