I am looking for examples (or, if it exists, a theory) of almost-monotone kernels. First, a bit of notation.

Recall that if $(\leq, \Omega)$ is a partially ordered set, then the set of measures $\mathcal{P}(\Omega)$ on $\Omega$ is partially ordered under:

$\mu \leq_{s} \nu$ if there exists a coupling of random variables $X,Y$ so that $X$ is distributed according to $\mu$, $Y$ is distributed according to $\nu$, and $X \leq Y$.

Finally, we say that a transition kernel $K: \Omega \rightarrow \mathcal{P}(\Omega) $ is monotone if, for all $\mu_{1}, \mu_{2} \in \mathcal{P}(\Omega)$ with $\mu_{1} \leq_{s} \mu_{2}$, we also have $\mu_{1} K \leq_{s} \mu_{2} K$.

I'm looking for any theory or examples related to "almost"-monotone kernels. Particularly of interest would be any results talking about "covering" (some subset of) stochastic kernels with small balls around monotone kernels.

A fantastic result might be: under some collection of conditions $X$, a kernel $K$ is always within $\epsilon = \epsilon(X)$ (as measured in some reasonable metric) of some kernel $K'$ that is monotone with respect to a `large' partial order. Even better would be some way to actually describe $K'$.

I am also extremely interested in any natural examples - families of "projections" from kernels to "nearby" monotone kernels.