That $Vol_{\omega}(\cdot)\leq Cap_{\omega}(\cdot)$ is a straightforward consequence of the definition (since $\omega_{0}=\omega$). It then follows from Stokes theorem that $\int_{X}\omega_{\varphi}^{n}=\int_{X}\omega^{n}$ for every $\varphi\in PSH(X,\omega)\cap L^{\infty}(X)$, thus $Vol_{\omega}(X)=Cap_{\omega}(X)$.

Property 2) is a straightforward consequence of the definitions.

If $\omega_{1}\leq\omega_{2}$ then $PSH(X,\omega_{1})\subset PSH(X,\omega_{2})$ hence $Cap_{\omega_{1}}(\cdot)\leq Cap_{\omega_{2}}(\cdot)$. Fix $A\geq 1$. If $\psi\in PSH(X,A\omega)$ is such that $0\leq\psi\leq 1$ then $\psi/A\in PSH(X,\omega)$ with $0\leq\psi/A\leq 1/A\leq 1$. Moreover $(A\omega+dd^{c}\psi)^{n}=A^{n}(\omega+dd^{c}(\psi/A))^{n}$. This shows $Cap_{A\omega}(\cdot)\leq A^{n}Cap_{\omega}(\cdot)$.

In particular if $\omega,\omega^{\prime}$ are both Kähler then $A^{-1}\omega\leq\omega^{\prime}\leq A\omega$ for some constant $A\geq 1$, hence $C^{-1}Cap_{\omega}(\cdot)\leq Cap_{\omega^{\prime}}(\cdot)\leq C\cdot Cap_{\omega}(\cdot)$ with $C=A^{n}$.

It remains to prove 4). It follows from the change of variables formula that if $\varphi\in PSH(X,\omega)$ with $0\leq\varphi\leq 1$ then

since $\varphi\circ f\in PSH(X,f^{*}\omega)$ with $0\leq\varphi\circ f\leq 1$. We infer $Cap_{\omega}(f(K))\leq Cap_{f^{*}\omega}(K)$. When $f$ is a $\omega$-isometry, i.e. $f\in Aut(X)$ with $f^{*}\omega=\omega$, then the mapping $\varphi\mapsto\varphi\circ f$ is an isomorphism of $\{u\in PSH(X,\omega)\,/\,0\leq u\leq 1\}$, whence $Cap_{\omega}(f(K))=Cap_{f^{*}\omega}(K)=Cap_{\omega}(K)$. ∎