Generalized Noise-Outsourcing Lemma for Borel data structures

Let $E$ and $D$ be Borel data structures, let $R$ denote the randomization Borel data structure with laws $\boldsymbol{\upsilon}$, and let $\operatorname{SYM}(E)$ denote the exchangeable probability laws on $E$. A natural transformation is understood componentwise over finite sets, and $\mu$-almost surely means that the relevant componentwise property holds for $\mu_a$-almost every $x\in E_a$. Generalized Noise-Outsourcing Lemma.

1. For every $\mu\in\operatorname{SYM}(E)$ and $\mu$-almost surely natural transformation $\eta:E\to\sP\circ D$, there exists a $\mu\otimes\boldsymbol{\upsilon}$-almost surely natural transformation $\widetilde\eta:E\times R\to D$ such that, for every finite set $a$,

$$\eta_a(x)=\boldsymbol{\upsilon}_a\circ\widetilde\eta_a(x,\mathord{\cdot})^{-1}$$

for $\mu_a$-almost all $x\in E_a$.

2. For every $\rho\in\operatorname{SYM}(E\times D)$ with first marginal $\mu\in\operatorname{SYM}(E)$, there exists a $\mu\otimes\boldsymbol{\upsilon}$-almost surely natural transformation $\eta:E\times R\to D$ such that

$$\rho=\mu\otimes\boldsymbol{\upsilon}\circ(1_E\otimes\eta)^{-1},$$

where $1_E\otimes\eta:E\times R\to E\times D$ has components $E_a\times R_a\to E_a\times D_a$ given by $(x,y)\mapsto(x,\eta_a(x,y))$.