Poisson matching lemma\]\[lem:cond\_phidiv\_gen\]Fix a distribution $P_{X,J,U,Y}$ and a probability kernel $Q_{U|Y}$, satisfying $J\in\mathbb{N}$ and $P_{U|X,J}(\cdot|X,J),Q_{U|Y}(\cdot|Y)\ll\mu$ almost surely. Let $(X,J)\sim P_{X,J}$, and $\{\bar{U}_{i},T_{i}\}_{i\in\mathbb{N}}$ be the points of a Poisson process with intensity measure $\mu\times\lambda_{\mathbb{R}_{\ge0}}$ independent of $(X,J)$. Let $U=\tilde{U}_{P_{U|X,J}(\cdot|X,J)}(J)$ and $Y|(X,J,U,\{\bar{U}_{i},T_{i}\}_{i})\sim P_{Y|X,J,U}(\cdot|X,J,U)$ (note that $(X,J,U,Y)\sim P_{X,J,U,Y}$ and $Y\leftrightarrow(X,J,U)\leftrightarrow\{\bar{U}_{i},T_{i}\}_{i}$). Then we have the following almost surely: $$\mathbf{E}\left[\left.\Upsilon_{P_{U|X,J}(\cdot|X,J)\Vert Q_{U|Y}(\cdot|Y)}(J)\,\right|\,X,J,U,Y\right]\le J\frac{dP_{U|X,J}(\cdot|X,J)}{dQ_{U|Y}(\cdot|Y)}(U)+1,$$ and for all $k\in\mathbb{N}$, $$\mathbf{P}\left\{ \left.\Upsilon_{P_{U|X,J}(\cdot|X,J)\Vert Q_{U|Y}(\cdot|Y)}(J)>k\,\right|\,X,J,U,Y\right\} \le\min\left\{ \frac{J}{k}\frac{dP_{U|X,J}(\cdot|X,J)}{dQ_{U|Y}(\cdot|Y)}(U),\,1\right\} ,$$ and $$\mathbf{P}\left\{ \left.\Upsilon_{P_{U|X,J}(\cdot|X,J)\Vert Q_{U|Y}(\cdot|Y)}(J)>1\,\right|\,X,J,U,Y\right\} \le1-\left(1-\min\left\{ \frac{dP_{U|X,J}(\cdot|X,J)}{dQ_{U|Y}(\cdot|Y)}(U),\,1\right\} \right)^{J}.$$ If $J=1$ almost surely, then we also have the following almost surely: for all $k\in\mathbb{N}$, $$\begin{aligned}
\mathbf{P}\left\{ \left.\Upsilon_{P_{U|X}(\cdot|X)\Vert Q_{U|Y}(\cdot|Y)}(1)>k\,\right|\,X,U,Y\right\}  & \le\Bigl(1-\Bigl(1+\frac{dP_{U|X}(\cdot|X)}{dQ_{U|Y}(\cdot|Y)}(U)\Bigr)^{-1}\Bigr)^{k}\\
 & \le1-\Bigl(1+k^{-1}\frac{dP_{U|X}(\cdot|X)}{dQ_{U|Y}(\cdot|Y)}(U)\Bigr)^{-1}.\end{aligned}$$

We can use the generalized Poisson matching lemma to extend Proposition \[prop:channel\] to the list decoding setting with fixed list size $\mathsf{J}$. The decoder outputs the list $\{\tilde{M}_{P_{X|Y}(\cdot|Y)\times P_{M}}(j)\}_{j\in[1:\mathsf{J}]}$. The error event becomes $(X,M)\notin\{(\tilde{X},\tilde{M})_{P_{X|Y}(\cdot|Y)\times P_{M}}(j)\}_{j\in[1:\mathsf{J}]}$. The probability of error is bounded by $\mathbf{E}\left[(1-(1+\mathsf{L}2^{-\iota_{X;Y}(X;Y)})^{-1})^{\mathsf{J}}\right]$.

One-shot Coding for Broadcast Channels and Mutual Covering
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The one-shot coding setting for the broadcast channel with common message is described as follows. Upon observing three independent messages $M_{j}\sim\mathrm{Unif}[1:\mathsf{L}_{j}]$, $j=0,1,2$, the encoder produces $X$, which is sent through the broadcast channel $P_{Y_{1},Y_{2}|X}$. Decoder $j$ observes $Y_{j}$ and recovers $\hat{M}_{0j}$ and $\hat{M}_{j}$ ($j=1,2$). The error probability is $P_{e}=\mathbf{P}\{(M_{0},M_{0},M_{1},M_{2})\neq(\hat{M}_{01},\hat{M}_{02},\hat{M}_{1},\hat{M}_{2})\}$.

We show 