5.27.37 Full cluster sets at the zero potentialC9D7C212A8BF

Proof. Choose relatively compact open sets $U_\ell $ exhausting $\Omega $, with $\overline U_\ell \subset U_{\ell +1}$. For each $\ell $, choose a countable uniformly dense subset of $C_0(U_{\ell +1};\mathbb R)$ consisting of smooth compactly supported functions, extended by zero to $\Omega $; such families exist because $C_0(U_{\ell +1};\mathbb R)$ is separable and its elements can be approximated uniformly by smooth functions. Add a nonnegative smooth compactly supported cutoff function equal to one near $\overline U_{\ell +1}$, and enumerate the union of all these functions as $(f_ i)$.

Call the maps $\alpha \mapsto \int _\Omega f_ i\, \mathrm d\alpha $ the coordinates. If positive measures converge to a fixed measure in every coordinate, the cutoff coordinates give common mass bounds on each $\overline U_{\ell +1}$, and uniform approximation then gives convergence against every $g\in C_ c(\Omega )$. The same argument, applied near a fixed measure, shows that $\alpha \mapsto \int _\Omega g\, \mathrm d\alpha $ is continuous for the topology generated by the coordinates: bound the relevant cutoff coordinate on a neighborhood, choose one smooth test that is uniformly close to $g$, and shrink the neighborhood using the coordinate of that test. Thus the coordinates generate exactly the vague topology, and they separate measures.

The function $s\mapsto s/(1+s)$ is increasing and subadditive on $[0,\infty )$. Applying it to the triangle inequalities for the coordinates gives the triangle inequality for $d_ v$, and $d_ v$ is positive on distinct measures because the coordinates separate them. Each coordinate is continuous for this metric, and the tails of the series are uniformly small, so the metric induces exactly the topology generated by the coordinates.

Finally, the finite atomic measures with nonnegative rational weights at points of a fixed countable dense subset of $\Omega $ form a countable dense family. To approximate a measure in finitely many coordinates, partition a compact neighborhood of the supports of the corresponding tests into sets on which all these tests have small oscillation, place an atom in each set of nonzero mass, and then perturb its location and weight. Only the finite mass of that compact neighborhood is used, and the remaining tail of the metric is arbitrarily small. $\square$

Proof. In a metric space, the full cluster set of a sequence is the intersection of the closures of its tails: a point of this intersection can be approximated within $1/k$ at an index larger than the preceding one, successively for $k=1,2,\ldots $. The full cluster set is therefore closed, and lemma 5.27.38 proves necessity.

For sufficiency, suppose first that $F$ is nonempty and closed. A subset of a separable metric space is separable, so we may choose a dense sequence $\mu _1,\mu _2,\ldots $ in $F$, repetitions allowed. Concatenating the finite lists $(\mu _1)$, $(\mu _1,\mu _2)$, $(\mu _1,\mu _2,\mu _3)$, and so on, we obtain a sequence $(\nu _ j)$ in which every $\mu _ k$ appears arbitrarily late. Hence every point of $F$ is the limit of a subsequence, and since $F$ is closed, no other subsequential limit occurs. If $F$ is empty, take $\nu _ j=j\delta _ a$ for a fixed $a\in \Omega $. A nonnegative compactly supported test function equal to one at $a$ has integral $j$ against $\nu _ j$, so this sequence has no vaguely convergent subsequence.

In either case, let $(f_ i)$ and $d_ v$ be as in lemma 5.27.38, and perform the finite atomic construction from the proof of theorem 5.27.36 separately for each $j$, with background zero, target $\nu _ j$ and tests $f_1,\ldots ,f_ j$. Thus we approximate $\nu _ j$ in these coordinates by a finite atomic measure with error less than $1/j$. If this measure has $N_ j>0$ atoms, apply proposition 5.27.35 successively with $t=\tau =(jN_ j)^{-1}$; if it has none, take $u_ j=0$. The resulting smooth bounded plurisubharmonic functions $u_ j\leq 0$ satisfy

On every fixed pair of concentric balls $B'\Subset B\Subset \Omega $, the bounds of that proposition for test measures $(\mathrm{dd}^{\mathrm{c}}w)^ n$, with $w\in \mathrm{PSH}(B)$ and $0\leq w\leq 1$, also sum to $C_{B',B,\Omega }/j$. Dividing by a positive threshold and taking the supremum over $w$ gives convergence in local capacity. These estimates require neither convergence of the targets $\nu _ j$ nor a common local mass bound for them.

Splitting the series defining the metric after its first $j$ terms gives

Consequently, the two sequences of measures have the same limits along corresponding subsequences, and hence the same full cluster set $F$. $\square$