Random Roth fusion lemma for generalized random rational functions

Let $K_1$, $K_2$, and $k$ be random compact subsets of $\mathbb C$ with $K_1(\omega)\cap K_2(\omega)=\emptyset$ for every $\omega$. Assume

$$|K_1(\Omega)|+|K_2(\Omega)|+|k(\Omega)|\leq\aleph_0.$$

Let $r_1,r_2:\Omega\times\mathbb C\to\overline{\mathbb C}$ be generalized random rational functions, with $r_i(\omega,\cdot)$ having no pole in $K_i(\omega)$ for $i=1,2$. If there is a positive error function $\varepsilon:\Omega\to\mathbb R^+$ such that

$$\lVert r_1(\omega,\cdot)-r_2(\omega,\cdot)\rVert_{k(\omega)}<\varepsilon(\omega),$$

then there exists a generalized random rational function $r:\Omega\times\mathbb C\to\overline{\mathbb C}$ and a measurable positive function $A:\Omega\to\mathbb R^+$, depending solely on $K_1$ and $K_2$, such that

$$\lVert r(\omega,\cdot)-r_1(\omega,\cdot)\rVert_{(K_1\cup k)(\omega)}<A(\omega)\varepsilon(\omega)$$

and

$$\lVert r(\omega,\cdot)-r_2(\omega,\cdot)\rVert_{(K_2\cup k)(\omega)}<A(\omega)\varepsilon(\omega)$$

for all $\omega\in\Omega$. Random Roth fusion conjecture. Under these hypotheses, the generalized random rational function $r$ and measurable positive function $A$ with the stated approximation bounds exist. The paper describes this as a random version of Roth's fusion lemma and states that it would follow directly from the random Runge conjecture above, which the paper does not prove.