Proposition 11.39.47DD0741AE5AA

Proof. By corollary 11.38.5, both duals are finite exactly on the probability measures of finite entropy; rescaling the reference volume to have mass one does not affect finiteness. A uniform bound for $|B_\varphi -B_\psi |$ passes to the defining suprema, so the supremum in the statement is at most $\Delta (\varphi ,\psi )$. Conversely, denote that supremum by $D$. If $\nu \in \mathcal A_\psi (G)$, then $\nu $ has finite entropy, and the dual minimum formula gives

Interchanging the references gives the opposite inequality, so $\Delta (\varphi ,\psi )\leq D$.

For bounded measurable $G$, take continuous approximations with the same uniform bound converging in $L^1(\mathrm{d}V)$; the continuity theorem theorem 11.33.5, applied to each of the two references, extends the scalar bound to $G$. Finally, a change of volume $\mathrm{d}V'=\mathrm{e}^ a\mathrm{d}V$ replaces each $B(G)$ by $B(G+a)$, and translation by $a\in C(X)$ is a bijection of the space of tests, so the supremum is unchanged. $\square$