It is enough to establish ${\mathcal{H}}(\alpha,A_{\alpha},\omega)$ for compact subsets, by regularity of $\mu$ and $Cap_{\omega}$. Let $K$ be a compact subset of $X$. It follows from Hölder’s inequality that

$$0\leq\mu(K)\leq||f||_{L^{p}(\omega^{n})}\left[\text{Vol}_{\omega}(K)\right]^{1/q},$$

where $1/p+1/q=1$. Note that $||f||_{L^{p}(\omega^{n})}=1$ since we assume $\mu$ is a probability measure. We claim that

(4)

$$\text{Vol}_{\omega}(K)\leq C_{\omega}\exp\left[-(Cap_{\omega}(K))^{-\gamma_{\omega}/n}\right],$$

for some constants $C_{\omega},\gamma_{\omega}>0$ that only depend on $\omega$. We will be done if we can prove (4) since we can then check by elementary computations that $\exp(-x^{-\delta})$ is dominated from above by $A_{\alpha}x^{\alpha}$, for all $x\in[0,1]$.

The set of functions ${\mathcal{F}}_{0}:=\{\varphi\in PSH(X,\omega)\,/\,\sup_{X}\varphi=0\}$ is compact in $L^{1}(X)$ (see proposition 2.7, [GZ 1]). These functions have Lelong numbers $\nu(\varphi,x)\leq\nu_{\omega}$ bounded from above by a uniform constant. It follows therefore from Skoda’s uniform integrability theorem [Z], that

$$\sup_{\varphi\in{\mathcal{F}}_{0}}\int\exp\left[-\frac{2\varphi}{\nu_{\omega}+1}\right]\omega^{n}\leq C_{2}<+\infty.$$

Set $\gamma_{\omega}:=2/(\nu_{\omega}+1)>0$ and let

$$V_{K,\omega}^{*}(x):=\left(\sup\{\varphi(x)\,/\,\varphi\in PSH(X,\omega),\,\varphi\leq 0\text{ on }K\}\right)^{*}$$

denote the Siciak extremal function of $K$ (see section 5.1 in [GZ 1]). Then

$$Vol_{\omega}(K)\leq\int_{X}\exp\left(-\gamma_{\omega}V_{K,\omega}^{*}\right)\omega^{n}\leq C_{2}T_{\omega}(K)^{\gamma_{\omega}},$$

where $T_{\omega}(K):=\exp(-\sup_{X}V_{K,\omega}^{*})$ denote the Alexander capacity of $K$ (see section 5.2 in [GZ 2]). It follows now from theorem 7.1 in [GZ 1] that

$$T_{\omega}(K)\leq e\exp\left[-Cap_{\omega}(K)^{-1/n}\right],$$

which yields (4). ∎