Pome

Corollary 16.9.336C532715D3CD

Corollary 16.9.33.6C532715D3CD Suppose that $(\varphi _ i)_{i\in I}$ is a net in $\mathrm{PSH}(X,\theta )$ and $\varphi \in \mathrm{PSH}(X,\theta )$. Then the following are equivalent:

  1. 28F7794ED0EE

    $\varphi _ i\xrightarrow {d_ S}\varphi $;

  2. CE229E338F57

    $\varphi _ i\lor \varphi \xrightarrow {d_ S}\varphi $ and

    F98DF8D2E20D 16.9.34
    \begin{equation} \lim _{i\in I}\int _ X \theta _{\varphi _ i}^ j\wedge \theta _{V_{\theta }}^{n-j}=\int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j} \end{equation}

    for each $j=0,\ldots ,n$;

  3. 90CB70DBE452

    for each $j=0,\ldots ,n$, (16.9.34) holds and

    217CFC062068 16.9.35
    \begin{equation} \lim _{i\in I}\int _ X \theta _{\varphi _ i\lor \varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}=\int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}. \end{equation}

Proof. The equivalence between 2 and 3 follows directly from 16.9.7.

1 $\implies $ 2. That $\varphi _ i\lor \varphi \xrightarrow {d_ S}\varphi $ follows from corollary 16.9.9. While (16.9.34) follows from theorem 16.9.30.

2 $\implies $ 1. By (16.9.10), we need to show that for each $j=0,\ldots ,n$, we have

\[ 2\int _ X \theta _{\varphi _ i\lor \varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{\varphi _ i}^ j\wedge \theta _{V_{\theta }}^{n-j}\to 0. \]

This follows from theorem 16.9.30 and (16.9.34). $\square$

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