Observation-Rate Family

Let $l$ and $\tilde{l}$ be natural numbers with $l\ge 2$ and $\tilde{l}\ge 1$, let $\Delta^l\subset\mathbb{R}^l$ be the probability simplex, and let $\tilde{B}$ be a nonnegative real number.

An observation-rate family on $l$ states with $\tilde{l}$ observation channels and rate bound $\tilde{B}$ is a family of functions

indexed by the pairs $(\sigma,\upsilon)$ with $\sigma\in\{1,\dots,l\}$ and $\upsilon\in\{1,\dots,\tilde{l}\}$, such that for every such pair:

1. (Bounds.) $0\le\tilde{\beta}(\sigma,\upsilon,\Sigma)\le\tilde{B}$ for all $\Sigma\in\Delta^l$.

2. (Continuity.) Whenever $(\Sigma_n)_{n\in\mathbb{N}}$ is a sequence in $\Delta^l$ such that the Euclidean distance $d(\Sigma_n,\Sigma)$ converges to $0$ for some $\Sigma\in\Delta^l$, then $\tilde{\beta}(\sigma,\upsilon,\Sigma_n)\to\tilde{\beta}(\sigma,\upsilon,\Sigma)$.