# How to characterize Radon Nikodym's derivative of a coupling with respect to any measure in the product space?

In my math essay of thesis I have defined the probability coupling as follows

$$\Pi(\mu,\nu)=\left\lbrace \pi \in \Omega \left\vert \begin{matrix} \pi(A\times\mathcal{Y})=\mu(A) \\ \pi(\mathcal{X} \times B)=\nu(B) \end{matrix}\right. \right\rbrace, \quad \begin{matrix} A\subset \mathcal{X} \\ B\subset \mathcal{Y} \end{matrix}$$

Where $\pi$ is a probability measure, $\Omega$ is the combined probability space and $\mathcal{X}$ and $\mathcal{Y}$ are the probability spaces of measures $\mu$ and $\nu$ respectfully. We suppose that $\mathcal{X}=\mathcal{Y}$.

We consider $\pi \in\Pi(\mu,\mathbb{P}_{N})$ where $\mathbb{P}_{N}$ is a empirical distribution, that is, there are $X_{1},\ldots,X_{N}$ such that $\mathbb{P}_{N}=\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}}$. Let $\mathbb{Q}\in \Omega$ be other measure such that $\pi$ is absolutelly continuous respect to $\mathbb{Q}$.

The question: We consider $\frac{d\pi}{d\mathbb{Q}}:\mathcal{X}\times \mathcal{Y}\rightarrow \mathbb{R}$ the Radon Nikodyn of $\pi$ respect to $\mathbb{Q}$, then for all $y\in\mathcal{X}$ we can define the function $\left.\frac{d\pi}{d\mathbb{Q}} \right|_{y}:\mathcal{X}\rightarrow \mathbb{R}$ defined by $\left.\frac{d\pi}{d\mathbb{Q}} \right|_{y}(x):=\frac{d\pi}{d\mathbb{Q}}(x,y)$.

Given that $\pi \in\Pi(\mu,\mathbb{P}_{N})$, is there a way to characterize $\left.\frac{d\pi}{d\mathbb{Q}} \right|_{X_{i}}$ for each $i=1,\ldots,N$?

If $\mathbb{Q}=m$ where $m$ is the Lebesgue measure on $\Omega$, is there such a characterization of $\left.\frac{d\pi}{dm} \right|_{X_{i}}$ for each $i=1,\ldots,N$?

You have too much notation here - which is quite confusing. For instance, what is your "combined probability space" $\Omega$ (it seems to be $X\times Y$ as this is where your Radon-Nikodym derivatives are defined) and how its elements can be measures? As far as I understand, you consider the joint distributions for which one marginal is $\mu$ and the other marginal is a discrete measure (let me denote it $\lambda$ rather than $\mathbb P_N$). These measures admit a very simple description (which consists just in looking at the conditional measures of the joint distribution conditioned by the points from $Y$), and which should imply answers to all your questions. Namely, any such measure can be presented as $$\sum_{y\in Y} \lambda(y) \mu_y \otimes\delta_y \;,$$ where $\mu_y$ are probability measures on $X$ such that $$\sum_y \lambda(y)\mu_y = \mu \;,$$ so that, in particular, $\mu_y$ are absolutely continuous with respect to $\mu$ and can be described by their densities.