Proposition 13.28.11.03E01F83CA7A For every $u\in \mathrm{PSH}(Y,\theta )$,
2C63A2647FAC
13.28.12
\begin{equation} \inf _{a\in \mathcal A_+(Y,\theta )}d_\theta (u,a) =d_\theta (u,P_\theta [u]_{\mathcal I})=\tau (u). \end{equation}
On ordinary models, $p\mapsto P_\theta [p]_{\mathcal I}$ is a $1$-Lipschitz retraction onto the ideal models, and $P_\theta [p]_{\mathcal I}$ is the unique ideal model nearest to $p$. There are $a_ j\in \mathcal A_+(Y,\theta )$ with $d_\theta (u,a_ j)\to 0$ if and only if $P_\theta [u]=P_\theta [u]_{\mathcal I}$.
Proof.
The ideal envelope of $u$ is $q_{D_ u}$, by lemma 20.3.20. For any positive atomic measure $E$ of mass at most $A$, the nonatomic measure $\sigma _ u$ and the signed atomic measure $D_ u-E$ are mutually singular. Thus (13.28.5) gives
57E1FB41A4EB
13.28.13
\begin{equation} d_\theta (u,q_ E)=\tau (u)+\| D_ u-E\| _{\mathrm{TV}}. \end{equation}
Among ideal models, the minimum is attained exactly at $E=D_ u$. Since the analytic models of positive mass are dense among the ideal models (corollary 13.28.7), this gives (13.28.12). Taking the atomic part does not increase total variation, which proves the Lipschitz assertion. Finally, lemma 13.28.1 shows that $P_\theta [u]=p_{\mu _ u}$ equals $q_{D_ u}=p_{D_ u}$ if and only if $\mu _ u=D_ u$, that is, $\tau (u)=0$.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.