Pairings into a Product, the Graph of a Measurable Map, and Couplings Concentrated on a Graph

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(Y,\mathcal{Y})$ be a measurable space and $(Z,d_{Z})$ a metric space with Borel $\sigma$-algebra $\mathcal{B}(Z)$, which is a $\sigma$-algebra by claim 2 of Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra, so that $(Z,\mathcal{B}(Z))$ is a measurable space and the product $\sigma$-algebras formed from it below are $\sigma$-algebras by the same claim. Let $Y\times Z$ carry the product $\sigma$-algebra $\mathcal{Y}\otimes\mathcal{B}(Z)$, with the coordinate projections $\mathrm{pr}_{Y}(y,z)=y$ and $\mathrm{pr}_{Z}(y,z)=z$, which are measurable by claim 5 of Sections of Product-Measurable Sets and Maps Are Measurable, and Insertion Maps into Products Are Measurable. Measurability of maps is that of Measurable Function and Real-Valued Measurable Function, and image measures are those of claim 1 of Image Measures, Measures with Densities, and Change of Variables. For maps $F:\Omega\to Y$ and $G:\Omega\to Z$ on a set $\Omega$, $(F,G):\Omega\to Y\times Z$ is the map $\omega\mapsto(F(\omega),G(\omega))$. For a map $S:Y\to Z$, $\mathrm{id}$ denotes the identity map of $Y$ and

$$\Gamma_{S}=\{(y,z)\in Y\times Z:\ z=S(y)\}$$

is the graph of $S$.

1. (Pairings)¶ For every measurable space $(\Omega,\mathcal{O})$ and all maps $F:\Omega\to Y$ and $G:\Omega\to Z$, measurable with respect to $\mathcal{O}$ and $\mathcal{Y}$, respectively $\mathcal{O}$ and $\mathcal{B}(Z)$, the map $(F,G)$ is measurable with respect to $\mathcal{O}$ and $\mathcal{Y}\otimes\mathcal{B}(Z)$. In particular, for every $S:Y\to Z$ measurable with respect to $\mathcal{Y}$ and $\mathcal{B}(Z)$, the map $(\mathrm{id},S)$ is measurable with respect to $\mathcal{Y}$ and $\mathcal{Y}\otimes\mathcal{B}(Z)$.

2. (The graph is measurable)¶ Suppose that $(Z,d_{Z})$ is separable, and let $S:Y\to Z$ be measurable with respect to $\mathcal{Y}$ and $\mathcal{B}(Z)$. Then $\Gamma_{S}\in\mathcal{Y}\otimes\mathcal{B}(Z)$.

3. (A coupling on the graph is the push-forward)¶ Suppose that $(Z,d_{Z})$ is separable, let $\mu$ be a probability measure on $(Y,\mathcal{Y})$, let $S:Y\to Z$ be measurable with respect to $\mathcal{Y}$ and $\mathcal{B}(Z)$, and let $\pi$ be a probability measure on $(Y\times Z,\mathcal{Y}\otimes\mathcal{B}(Z))$ with $(\mathrm{pr}_{Y})_{\#}\pi=\mu$ and $\pi(\Gamma_{S})=1$, the number $\pi(\Gamma_{S})$ being defined by claim 2. Then $\pi=(\mathrm{id},S)_{\#}\mu$, an image measure defined by claim 1, and $(\mathrm{pr}_{Z})_{\#}\pi=S_{\#}\mu$.