1) For all Borel subsets $K^{\prime}\subset K\subset X$, $T_{\omega}(K^{\prime})\leq T_{\omega}(K)\leq T_{\omega}(X)=1$.

2) If $\omega_{1}\leq\omega_{2}$ then $T_{\omega_{1}}(\cdot)\geq T_{\omega_{2}}(\cdot)$. Forall $A>0$, $T_{A\omega}(\cdot)=[T_{\omega}(\cdot)]^{A}$. In particular if $\omega$ and $\omega^{\prime}$ are both Kähler then there exists $C\geq 1$ such that

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

where $C=\exp(\sup_{X}\chi-\inf_{X}\chi)\geq 1$.

4) If $f:X\rightarrow X$ is a holomorphic map then $T_{f^{*}\omega}(\cdot)\leq T_{\omega}\circ f(\cdot)$. In particular if $f$ is a $\omega$-isometry then $T_{\omega}\circ f=T_{\omega}$.