Corollary 16.9.13.D47345F0C3C8 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Then
\[ \left| \int _ X \theta _{\varphi }^ n-\int _ X \theta _{\psi }^ n \right|\leq D_ n d_ S(\varphi ,\psi ), \]
where $D_ n=3(n+1)C_ n$ with $C_ n$ being the same constant as in 16.9.4.
Proof.
We compute
\[ \begin{aligned} \left| \int _ X \theta _{\varphi }^ n-\int _ X \theta _{\psi }^ n \right|\leq & \left|2\int _ X \theta _{\varphi \lor \psi }^ n-\int _ X \theta _{\varphi }^ n-\int _ X \theta _{\psi }^ n\right| + 2\left| \int _ X \theta _{\varphi \lor \psi }^ n-\int _ X \theta _{\varphi }^ n\right|\\ \leq & (n+1)C_ n d_ S(\varphi ,\psi )+2(n+1) d_ S(\varphi ,\varphi \lor \psi )\\ \leq & (n+1)C_ n d_ S(\varphi ,\psi )+2(n+1)C_ n d_ S(\varphi ,\psi ), \end{aligned} \]
where the first line is just the triangle inequality, the second line follows from corollary 16.9.9 and the third line follows from 16.9.4.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.