Proposition 3.4 . [033I]

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Proposition 3.4.

Let $K$ be a Borel subset of $X$ .

1) If $K^{\prime}\subset K$ then $V_{X,\omega}\leq V_{K,\omega}\leq V_{K^{\prime},\omega}$ and $\sup_{X}V_{X,\omega}=0$ . Furthermore $V_{X,\omega}\equiv 0$ when $\omega\geq 0$ .

2) If $\omega_{1}\leq\omega_{2}$ then $V_{K,\omega_{1}}\leq V_{K,\omega_{2}}$ .

3) For all $A>0$ , $V_{K,A\omega}=A\cdot V_{K,\omega}$ .

4) If $\omega^{\prime}=\omega+dd^{c}\chi$ then

-\chi+\inf_{X}\chi+V_{K,\omega}\leq V_{K,\omega^{\prime}}\leq V_{K,\omega}+\sup_{X}\chi-\chi.

5) If $f:X\rightarrow X$ is holomorphic then

V_{f(K),\omega}\circ f\leq V_{K,f^{*}\omega}.

In particular if $f$ is a $\omega$ -isometry then $V_{f(K),\omega}=V_{K,\omega}$ .