Let $(\Xi,\mathcal{E})$ be a measurable space and $\xi$ and $\xi'$ random variables with distributions $\mu$ and $\vartheta$ respectively in this space.
We say that the measure $\Pi$ in $\Xi^{2}$ is the coupling of measures $\mu$ and $\vartheta$ if $\Pi$ is a joint distribution of $\xi$ and $\xi'$ with marginals $\mu$ and $\vartheta$ respectively.
From the law of total probability we can infer the following:
If we suppose that $\xi'$ take only the values $\widehat{\xi}_{i}\in \Xi$ for $i=1,2,\ldots, N$ and ${\displaystyle \vartheta=\frac{1}{N}\sum_{i=1}^{N}\delta_{\widehat{\xi}_{i}} }$, then by the law of total probability follows $$\Pi=\frac{1}{N}\sum_{i=1}^{N}\delta_{\widehat{\xi}_{i}}\otimes \mathbb{Q}_{i} \tag{1}$$ where $\mathbb{Q}_{i}$ is the conditional distribution of $\xi$ given $\xi'=\widehat{\xi}_{i}$.
My Question This doubt is born because I am reading an article in this link in pag 12 which use the fact (1). In short, what I do not understand is how from (1) we can infer the following:
$$\int_{\Xi^{2}}\left\|\xi-\xi'\right\|\Pi(d\xi,d\xi')=\frac{1}{N}\sum_{i=1}^{N}\int_{\Xi}\left\|\xi-\widehat{\xi}_{i}\right\|\mathbb{Q}_{i}(d\xi). \tag{2}$$
Remark: I think that (2) it may be due to some characterization of the integral with respect to the product measure, but I do not know what that characterization is, so I ask this community to help with any suggestion or solution.
This is due to Fubbini's Theorem as follows: \begin{align} \int_{\Xi^{2}}\left\|\xi-\xi' \right\|\Pi(d\xi,d\xi') &= \int_{\Xi^{2}}\left\|\xi-\xi'\right\|\frac{1}{N} \sum_{i=1}^{N}\delta_{\widehat{\xi}_{i}}\otimes\mathbb{Q}_{i}(d\xi',d\xi) \nonumber \\ &= \frac{1}{N}\sum_{i=1}^{N}\int_{\Xi^{2}}\left\|\xi-\xi'\right\|\delta_{\widehat{\xi}_{i}}\otimes\mathbb{Q}_{i}(d\xi',d\xi) \nonumber \\ &= \frac{1}{N}\sum_{i=1}^{N}\int_{\Xi}\left(\int_{\Xi}\left\|\xi-\xi'\right\| \delta_{\widehat{\xi}_{i}}(d\xi') \right)\mathbb{Q}_{i}(d\xi) \nonumber \\ &= \frac{1}{N}\sum_{i=1}^{N}\int_{\Xi} \left\|\xi-\widehat{\xi}_{i}\right\| \mathbb{Q}_{i}(d\xi) \nonumber \end{align}