Zero-cost information capacity of a countable-input finite-output DMC

A countable-input finite-output DMC has input alphabet ${\mathbb{N}}_{>0}=\{1,2,\dots \}$, a finite nonempty output alphabet $\mathcal{Y}$, and row probabilities $W(y\mid x)\ge 0$ with ${\sum }_{y\in \mathcal{Y}}W(y\mid x)=1$ for each $x$. At blocklength $n$, its channel law is $W^n(y^n\mid x^n)={\prod }_{t=1}^{n}W(y_t\mid x_t)$. An input law is a sequence $(p_x{)}_{x\ge 1}$ with $p_x\ge 0$ and ${\sum }_x{p}_x=1$. Give every input symbol cost $c(x)=0$ and budget $\mathrm{\Gamma }=0$. Its output law is $q(y)={\sum }_x{p}_{x}W(y\mid x)$, and define $H(q)=-{\sum }_{y\in \mathcal{Y}}q(y)\ln q(y),H(Y\mid X)={\sum }_{x\ge 1}{p}_x[-{\sum }_{y\in \mathcal{Y}}W(y\mid x)\ln W(y\mid x)],$ where finite entropy is as in Finite distributions, entropy, divergence, and stochastic matrices, $0\ln 0=0$, and logarithms are natural The natural logarithm as the inverse of the exponential function. The conditional-entropy series converges because each summand entropy lies between $0$ and $\ln \mathrm{\mid }\mathcal{Y}\mathrm{\mid }$ by Finite-simplex entropy is continuous and concave. Define $I(p;W):=H(q)-H(Y\mid X)$ and the zero-cost information capacity $C_I(0):={\sup }_{p}I(p;W)$ over all countable input laws. This definition alone makes no assertion that the supremum is attained or equals an operational coding capacity.