Corollary 20.3.22D416A3AB0845

Proof. By lemma 20.3.20, the space corresponds to the positive atomic measures of mass at most $A-\varepsilon $. This set is complete for total variation. Indeed, a Cauchy sequence has as limit a positive finite measure with the same mass bound. The union of the countable supports of the sequence is countable, and total-variation convergence shows that the limit gives zero mass to the complement of that union, so the limit is again atomic. The two metric estimates of theorem 20.3.21 transfer completeness to the ideal envelopes.

Finite truncation, followed by rational approximation of each coefficient from below, preserves the mass bound and converges in total variation. The resulting Green potentials have finitely many logarithmic poles with rational coefficients, and their ideal envelopes have the same ordinary types by lemma 20.3.20. These envelopes therefore have analytic singularities, and the upper metric estimate proves density.

Finally, fix $0<a<A-\varepsilon $. The measures $a\delta _ p$, for distinct $p\in Y$, have pairwise total-variation distance $2a$, so their ideal envelopes have pairwise $d_ S$ distance at least $a/48$. This infinite separated family rules out total boundedness. $\square$