Gluing two Euclidean couplings

Apply [F2] to $\alpha$ with its second coordinate y as conditioning variable, obtaining a probability kernel $K_y(dx)$ with $\alpha (A\times B)={\int }_B{K}_y(A)d\rho (y)$. The supplier conditions on the second marginal, so for $\beta$ first define the coordinate swap $S(y,z)=(z,y)$ and $\widehat{\beta }=S_{\mathrm{\#}}\beta$ on ${\mathbb{R}}^{d_3}\times {\mathbb{R}}^{d_2}$. By [F7], this is a Borel probability whose second marginal is the common law $\rho$. Apply [F2] to $\widehat{\beta }$ to obtain a probability kernel $L_y(dz)$ with $\widehat{\beta }(C\times B)={\int }_B{L}_y(C)d\rho (y)$, equivalently $\beta (B\times C)={\int }_B{L}_y(C)d\rho (y)$. Thus for every Borel $A\subseteq {\mathbb{R}}^{d_1}$, $B\subseteq {\mathbb{R}}^{d_2}$, and $C\subseteq {\mathbb{R}}^{d_3}$ both joint laws are disintegrated over the same conditioning law.

For each $y$, let $Q_y=K_y\otimes L_y$, the product probability on ${\mathbb{R}}^{d_1}\times {\mathbb{R}}^{d_3}$. For a rectangle $A\times C$, $Q_y(A\times C)=K_y(A)L_y(C)$ is measurable in $y$. The sets $D$ for which $y\mapsto Q_y(D)$ is measurable form a lambda-system: the whole space has value one, complements have value $1-Q_y(D)$, and disjoint unions have value the increasing limit of finite sums. Rectangles form a pi-system generating the product Borel sigma-algebra by [F1] and [F4]. Thus [F4] proves measurability for every Borel $D$. The same lambda-system argument applied to rectangles $B\times A\times C$ in the variables $(y,x,z)$ shows that $y\mapsto Q_y(E_y)$ is measurable for each Borel $E\subseteq {\mathbb{R}}^{d_2}\times {\mathbb{R}}^{d_1}\times {\mathbb{R}}^{d_3}$, where $E_y=\{(x,z):(y,x,z)\in E\}$.

Define $\tilde{\gamma }(E)=\int Q_y(E_y)d\rho (y)$ on the product Borel sigma-algebra in the coordinate order $(y,x,z)$. It is nonnegative and has total mass one. For disjoint $E_j$, their sections are disjoint, so countable additivity of $Q_y$ followed by monotone convergence [F6] gives $\tilde{\gamma }({\bigcup }_j{E}_j)={\sum }_j\tilde{\gamma }(E_j)$. Hence $\tilde{\gamma }$ is a Borel probability.

For Borel rectangles $B\times A\times {\mathbb{R}}^{d_3}$ and $B\times {\mathbb{R}}^{d_1}\times C$, the definition and the two identities in step 1.1 give respectively $\tilde{\gamma }(B\times A\times {\mathbb{R}}^{d_3})={\int }_B{K}_y(A)L_y({\mathbb{R}}^{d_3})d\rho (y)=\alpha (A\times B)$ and $\tilde{\gamma }(B\times {\mathbb{R}}^{d_1}\times C)={\int }_B{L}_y(C)K_y({\mathbb{R}}^{d_1})d\rho (y)=\beta (B\times C)$. Rectangles determine finite Borel measures by [F4], so these identities identify the two pair marginals in coordinate order $(y,x,z)$. Define the Borel coordinate permutation $P(y,x,z)=(x,y,z)$ and set $\gamma =P_{\mathrm{\#}}\tilde{\gamma }$, meaning $\gamma (H)=\tilde{\gamma }(P^{-1}(H))$ for each Borel $H$; by [F7] this is a Borel probability. The coordinate projections satisfy ${\mathrm{pr}}_{x,y}\circ P(y,x,z)=(x,y)$ and ${\mathrm{pr}}_{y,z}\circ P(y,x,z)=(y,z)$, hence their pushforwards are exactly $\alpha$ and $\beta$.