Let $u,u^{\prime}\in C^{0}(X)$.

• (i)

  $P_{\theta}(u)$ is $\theta$-psh, and is the largest $\theta$-psh function dominated by $u$ on $X$.

• (ii)

  $P_{\theta}$ is non-decreasing, i.e. $u\leq v\Rightarrow P_{\theta}(u)\leq P_{\theta}(v)$.

• (iii)

  $P_{\theta}(u)$ is concave in both arguments, i.e.

  $$P_{t\theta+(1-t)\theta^{\prime}}\left(tu+(1-t)u^{\prime}\right)\geq tP_{\theta}(u)+(1-t)P_{\theta}^{\prime}(u^{\prime})$$

  for $0\leq t\leq 1$.

• (iv)

  For each $c\in\mathbf{R}$ we have $P_{\theta}(u+c)=P_{\theta}(u)+c$.

• (v)

  For each $v\in\mathcal{D}(X)$ we have $P_{\theta}(u)=P_{\theta+dd^{c}v}(u-v)+v$.

• (vi)

  $P_{\theta}$ is $1$-Lipschitz continuous with respect to the sup-norm, i.e. $\sup_{X}|P_{\theta}(u)-P_{\theta}(v)|\leq\sup_{X}|u-v|$.

• (vii)

  Given a determination $\mathcal{X}$ of $\theta$ and a convergent sequence $\theta_{m}\to\theta$ in $N^{1}(\mathcal{X}/S)$, we have $P_{\theta_{m}}(u)\to P_{\theta}(u)$ uniformly on $X$.