Definition 9.2 — Dobrushin coefficient

For arbitrary types $X$ and $Y$, each equipped with a measurable-space structure, and any kernel $K$ from $X$ to $Y$ satisfying the Markov-kernel assumption (so $K(x)$ is a probability measure on $Y$ for every $x\in X$), the definition assigns the real number $\delta(K)=\sup\{d(K(x),K(y)):x,y\in X\}$, where, for any two measures $\mu,\nu$ on a measurable space, the imported definition is $d(\mu,\nu)=\sup\{|\operatorname{toReal}(\mu(A))-\operatorname{toReal}(\nu(A))|:A\text{ is a measurable subset}\}$. Here the set $A$ ranges over measurable subsets of $Y$ when evaluating $d(K(x),K(y))$, and $\operatorname{toReal}$ converts a finite extended nonnegative real to its real value and sends $+\infty$ to $0$. Both suprema are suprema in $\mathbb R$ of sets of real numbers; the real supremum convention gives $0$ for an empty set or a set unbounded above. For these probability measures the inner set contains $0$ (take $A=\varnothing$) and is bounded above by $1$, and the outer set is also bounded above by $1$. The pair $x,y$ may consist of equal points; there is no distinctness or nonemptiness hypothesis on $X$ or $Y$. If $X$ is empty the outer set is empty and $\delta(K)=0$; if $X$ is nonempty the outer set contains $0$. An empty $Y$ admits no such Markov kernel when $X$ is nonempty, since there is no probability measure on the empty space. No input measure on $X$ is an argument of this definition.