Corollary 13.28.7FB89848D2500

Proof. By lemma 13.28.1, theorem 13.28.4, the space of ordinary models is isometric to the set of polar measures of mass at most $A$ with the total variation norm. This set is closed in total variation: a limit is positive, satisfies the mass bound, and is carried by the countable union of polar carriers of the sequence. Completeness therefore follows from that of finite signed measures. For two such measures $\mu ,\nu $, the measures $\mu _ t=(1-t)\mu +t\nu $, $0\leq t\leq 1$, remain in this set and satisfy

Their models give a constant-speed geodesic.

The atomic measures form a closed convex subset, since a total-variation limit is carried by the countable union of the sets of atoms of the sequence. Restricting to finitely many atoms and approximating the coefficients from below by rational numbers, with an arbitrarily small further decrease if needed, approximates every such measure by finite atomic measures with rational coefficients and mass strictly less than $A$. Their models have analytic singularities by lemma 20.3.20, and (13.28.6) gives the density. $\square$