for the sake of completeness.

\[Conditional Poisson matching lemma\]\[lem:cond\_phidiv\]Fix a distribution $P_{X,U,Y}$ and a probability kernel $Q_{U|Y}$ (that is not necessarily $P_{U|Y}$) satisfying $P_{U|X}(\cdot|X),Q_{U|Y}(\cdot|Y)\ll\mu$ almost surely. Let $X\sim P_{X}$, 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 $X$. Let $U=\tilde{U}_{P_{U|X}(\cdot|X)}$ and $Y|(X,U,\{\bar{U}_{i},T_{i}\}_{i})\sim P_{Y|X,U}(\cdot|X,U)$ (note that $(X,U,Y)\sim P_{X,U,Y}$ and $Y\leftrightarrow(X,U)\leftrightarrow\{\bar{U}_{i},T_{i}\}_{i}$). Then we have the following almost surely: $$\mathbf{P}\left\{ \left.\tilde{U}_{Q_{U|Y}(\cdot|Y)}\neq U\,\right|\,X,U,Y\right\} \le1-\left(1+\frac{dP_{U|X}(\cdot|X)}{dQ_{U|Y}(\cdot|Y)}(U)\right)^{-1}.$$

The condition that $P_{U|X}(\cdot|X),Q_{U|Y}(\cdot|Y)\ll\mu$ almost surely is satisfied, for example, when $\mu=P_{U}$, $Q_{U|Y}=P_{U|Y}$, $P_{UX}\ll P_{U}\times P_{X}$ and $P_{UY}\ll P_{U}\times P_{Y}$. Note that since $X{\perp\!\!\!\perp}\{\bar{U}_{i},T_{i}\}_{i}$, we have $\tilde{U}_{P_{U|X}(\cdot|X)}|X\sim P_{U|X}$, whereas $Y$ may not be independent of $\{\bar{U}_{i},T_{i}\}_{i}$, so $\tilde{U}_{Q_{U|Y}(\cdot|Y)}$ may not follow the conditional distribution $Q_{U|Y}$.

One-shot Channel Coding\[sec:channel\]
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To demonstrate the application of the Poisson matching lemma, we apply it to recover a bound for one-shot channel coding in [@yassaee2013oneshot] (with a slight penalty of having $\mathsf{L}$ instead of $\mathsf{L}-1$). Upon observing $M\sim\mathrm{Unif}[1:\mathsf{L}]$, the encoder produces $X$, which is sent through the channel $P_{Y|X}$. The decoder observes $Y$ and recovers $\hat{M}$ with error probability $P_{e}=\mathbf{P}\{M\neq\hat{M}\}$.

\[prop:channel\]Fix any $P_{X}$. There exists a code for the channel $P_{Y|X}$, with message $M\sim\mathrm{Unif}[1:\mathsf{L}]$, with average error probability $$P_{e}\le\mathbf{E}\left[1-\left(1+\mathsf{L}2^{-\iota_{X;Y}(X;Y)}\right)^{-1}\right]$$ if $P_{XY}\ll P_{X}\times P_{Y}$.

Let $\{(\bar{X}_{i},\bar{M}_{i}),T_{i}\}_{i\in\mathbb{N}}$ be the 