Let $(X, \leq)$ be a partial order and $\Sigma\_X$ a $\sigma$-algebra on $X$. Is the set $\{(x, y) \in X\times X \mid x \leq y\}$ measurable with respect to the product $\sigma$-algebra? 



Ordered measurable spaces
This can fail, even if you assume that the lower cones of the order are measurable with respect to the algebra $\Sigma$. For example, consider the real numbers $X=\mathbb{R}$ and let $\Sigma$ be the usual Borel algebra. Let's assume the continuum hypothesis, and let $\leq$ be a well-order of the reals in order type $\omega\_1$. Thus, every lower cone $\{x\mid x\leq y\}$ is a countable set, which is therefore Borel. But the graph of the relation $\{(x,y)\mid x\leq y\}$ is not measurable, since all its horizontal sections are countable, but all its vertical sections are co-countable, which would violate Fubini's theorem.


