Just to double check: The case $i=j$ should really be included in the sum in the definition of $p$?

@Ano2Math5: In the version of the book which is linked, one has "Lemma 3.1 (kernel criteria)", where Chapter 3 is "Kernels, Disintegration, and Invariance". Maybe you have a different version? Lemma 1.14 on the other hand is "Lemma 1.14 (functional representation, Doob)"

I see. I haven't seen this kind of problem. It seems slightly related to weak optimal transport (see the paper "Kantorovich duality for general transport costs and applications" by Gozlan et al), where the cost depends on $\gamma \mid a$. In contrast to your problem, one still integrates over $\alpha$. So weak OT are problems of the form $\inf_{\gamma \in \Gamma(\alpha, \nu)} \int C(a, \gamma \mid a) \alpha(da)$ (choosing $C(a, \gamma \mid a) = \int c(a, b) \gamma(db \mid a)$ reduces to normal OT).

I'm trying to understand the problem a bit better: It seems the value $V_a(\alpha, \nu)$ does not really depend on $\alpha$, but only on $\alpha(a)$? Also, if I don't miss anything, the solution seems quite simple: Basically $\gamma \mid a$ fills the part of $\nu$ where $u$ has the largest values. This means $\gamma \mid a$ simply has to equal the (scaled) restriction of $\nu$ to a set $A$, where the set $A$ is such that $\nu(A) = \alpha(a)$ and $u(x) \geq u(y)$ for all $x \in A \cap supp(\nu), y \in A^C \cap supp(\nu)$.

Note that $\rho(x, F) - \rho(y, F) \leq \sup_{z \in F} (\rho(x, z) - \rho(y, z)) \leq \rho(x, y)$.

Could you clarify what you mean by "construct" (beyond proving that such a probability space exists)?

Do I understand correctly that $\mu_2$ is random, and you have one fixed distribution, $\mathcal{N}(\mu_1, \mathbb{I}_d)$ and one random distribution, $\mathcal{N}(\mu_2(\omega), \mathbb{I}_d)$. The KL divergence takes as input two fixed distributions. Are you interested in the expected KL divergence (expectation taken over the distribution of $\mu_2$), or something else?

Two brief comments: First, it seems if $S \subset S'$ and $S'$ is $f$-cyclically monotone, then $S$ is also $f$-cyclically monotone, so why include the $S'$ in your statement? And second, I think there may be some hope if you restrict to twice continuously differentiable $f$ and $g$ with some non-degeneracy assumptions using similar ideas to the proof of Theorem 1.2 of McCann, Pass and Warren "Rectifiability of Optimal Transportation Plans".

Do we know if the supremum is always finite? In particular, if the supremum were infinite for circles, then it would also be infinite for large enough squares using the same strategy.

Not sure if that is your question, but of course under sufficient moments ($q$-th moments for $q > 2p$) you have the same behavior as for the uniform distribution, which is shown in the paper by Fournier and Guillin "On the rate of convergence in Wasserstein distance of the empirical measure."

The last two integrals simply converge to zero. I think that there is a lot of slack in the statement of the Lemma by not specifying $c_1$. For the term $2^{-1/\beta}$, I think any other constant $(1+\delta)$ instead of 2 would work as well where for arbitrarily small $c_1$, the $\delta$ also can be chosen smaller. The paper seems to side-step this precise weighing and simply takes $\delta=1$.