Mittag-Leffler on plane domains

Choose an increasing compact exhaustion $C_1\subseteq C_2\subseteq \cdots$ of $\mathrm{\Omega }$ with $C_n\subseteq C_{n+1}^{\circ }$ and ${\bigcup }_n{C}_n=\mathrm{\Omega }$. [given, L1, L3, construct] Recursively apply [L1] to choose compact Jordan neighbourhoods of the $C_n$ and then fill every complementary component lying entirely in $\mathrm{\Omega }$. This gives an increasing exhaustion $K_1\subseteq K_2\subseteq \cdots$ of compact subsets of $\mathrm{\Omega }$ such that every connected component of $\widehat{\mathbb{C}}\setminus K_n$ meets $\widehat{\mathbb{C}}\setminus \mathrm{\Omega }$. Because $A$ is discrete, each set $A_n:=A\cap (K_n\setminus K_{n-1})(K_0:=\mathrm{\varnothing })$ is finite. Let $f_n(z):={\sum }_{a\in A_n}{p}_a(z)\mathrm{.}$ Then $f_n$ is meromorphic on $\mathrm{\Omega }$ and holomorphic on a neighbourhood of $K_{n-1}$.

Fix a set $P\subseteq \widehat{\mathbb{C}}\setminus \mathrm{\Omega }$ meeting every connected component of $\widehat{\mathbb{C}}\setminus \mathrm{\Omega }$. [L2, step 1.1, choose] Step 1.1 makes $P$ meet every connected component of $\widehat{\mathbb{C}}\setminus K_{n-1}$ as well. Put $r_1:=0$ and $h_1:=f_1$. For each $n\ge 2$, apply [L2] to $f_n$ on a neighbourhood of $K_{n-1}$ and choose a rational function $r_n$ with poles in $P$ such that ${\sup }_{K_{n-1}}\mathrm{\mid }{f}_n-r_n\mathrm{\mid }<2^{-n}\mathrm{.}$ Then $h_n:=f_n-r_n$ is meromorphic on $\mathrm{\Omega }$, has the same principal parts as $f_n$ on $A_n$, and is uniformly small on $K_{n-1}$.

For a compact set $K\subseteq \mathrm{\Omega }\setminus A$, choose $N$ with [step 2.1, algebra, discharge-construct] $K\subseteq K_{N-1}$. Then for every $n\ge N$, step 2.1 gives ${\sup }_K\mathrm{\mid }{h}_n\mathrm{\mid }\le 2^{-n}$. Therefore ${\sum }_n{h}_n$ converges uniformly on $K$. Only finitely many layers $A_n$ meet a given compact set, so $f:={\sum }_{n\ge 1}{h}_n$ is meromorphic on $\mathrm{\Omega }$, and its principal part at each $a\in A_n$ is exactly $p_a$.