Definition 9.1 — strong total-variation coefficient

For arbitrary types $X$ and $Y$, each equipped with a measurable space, and a Markov kernel $K$ from $X$ to $Y$ (so $K(x)$ is a probability measure on $Y$ for every $x$, and its evaluations on measurable sets vary measurably with $x$), the definition assigns a real number $c(K)$ as follows. For probability measures $\mu,\nu$ on $X$, let $\mu K$ be the probability measure whose value on a measurable set $B\subseteq Y$ is $\int_X K(x)(B)\,d\mu(x)$, and define $\nu K$ likewise. For measures $\alpha,\beta$ on a measurable space $Z$, put $D_Z(\alpha,\beta)=\sup_{A\subseteq Z,\ A\text{ measurable}} |\operatorname{toReal}(\alpha(A))-\operatorname{toReal}(\beta(A))|$, where the supremum is taken in $\mathbb R$ and $\operatorname{toReal}$ converts an extended nonnegative real to a real, sending $+\infty$ to $0$. Then $c(K)=\sup\{D_Y(\mu K,\nu K)/D_X(\mu,\nu):\mu,\nu\text{ are probability measures on }X,\ \mu\ne\nu\}$, again a supremum in $\mathbb R$. Thus the outer set consists of all real ratios witnessed by two distinct probability measures, with no additional hypotheses on $X$, $Y$, or their measurable spaces; in particular the quantifiers do not require the underlying types to be nonempty. These are real suprema, with the convention that the supremum of the empty set, or of a set not bounded above, is $0$. In particular, if $X$ admits no pair of distinct probability measures, the outer set is empty and $c(K)=0$. All measures evaluated in these ratios are probability measures, so their set evaluations are finite; distinct probability measures give $D_X(\mu,\nu)>0$. The division is real division (whose total definition gives $a/0=0$, although a zero denominator does not occur for an admitted pair).