Theorem 23.1.628BCBC6CB58C6

Theorem 23.1.62.8BCBC6CB58C6 Within $\mathcal R$ the following assertions hold.

1. Every individual $\nu _{W,u}$ is finite if and only if $\sup _{t\gg 1}\sqrt t\, W(t)<\infty $. This is also equivalent to a uniform total-mass bound. In this case

   \[ \sup _{u\in \mathcal R}\nu _{W,u}(X)\leq 12Q(W). \]

   If the condition fails there is one $u\in \mathcal R$ with infinite measure.

2. Uniform deep tails hold if and only if $\sqrt t\, W(t)\to 0$. For all $u\in \mathcal R$ and $T\geq 0$,

   \[ \nu _{W,u}(\{ -u>T\} )\leq 12Q_ T(W). \]

   The inequality is understood in the extended sense.

3. Uniform absolute continuity relative to $\operatorname {Cap}_\omega $ holds if and only if $\sqrt t\, W(t)\to 0$.

4. The assignment $u\mapsto \nu _{W,u}$ is a map into finite measures and is continuous from capacity convergence in $\mathcal R$ to weak measure convergence if and only if $\sqrt t\, W(t)\to 0$. When $\sqrt t\, W(t)$ is bounded but does not tend to zero, there are smooth $u_ j\in \mathcal R$ converging in capacity to $g\in \mathcal R$, whose weighted measures have a weak cluster measure charging the pole divisor of $g$.