It is clear that $(1)\Rightarrow(2)\Rightarrow(3)$. The implication $(3)\Rightarrow(2)$ follows from the equality $P_{\theta}(f)-f=P_{\theta+dd^{c}f}(0)$. It remains to prove $(2)\Rightarrow(1)$. We may write a given $f\in C^{0}(X)$ as a uniform limit on $X$ of model functions $f_{j}$, and $P_{\theta}(f_{j})\to P_{\theta}(f)$ uniformly on $X$ thanks to the Lipschitz property of $P_{\theta}$, see Proposition 2.15. By Theorem 3.1 we thus have

$$\left(\theta+dd^{c}P_{\theta}(f_{j})\right)^{n}\to\left(\theta+dd^{c}P_{\theta}(f)\right)^{n}$$

in the weak topology of measures. Since $P_{\theta}(f_{j})-f_{j}\to P_{\theta}(f)-f$ uniformly on $X$ and the measures $(\theta+dd^{c}P_{\theta}(f_{j}))^{n}$ have uniformly bounded (in fact, constant) mass, it follows that

$$\int\left(P_{\theta}(f)-f\right)\left(\theta+dd^{c}\ P_{\theta}(f)\right)^{n}=\lim_{j}\int\left(P_{\theta}(f_{j})-f_{j}\right)\left(\theta+dd^{c}\ P_{\theta}(f_{j})\right)^{n}=0.$$

Let us prove the final assertion. Pick $\theta^{\prime}\in\mathcal{Z}^{1,1}(X)$ such that $\{\theta^{\prime}\}=\{\theta\}$ in $N^{1}(X)$. By the analogue of the $dd^{c}$-lemma proved in [BFJ11, Theorem 4.3] there exists $g\in\mathcal{D}(X)$ such that $\theta^{\prime}=\theta+dd^{c}g$. Observe that a function $\varphi$ is $\theta^{\prime}$-psh iff $\varphi+g$ is $\theta$-psh. As a consequence we get $P_{\theta^{\prime}}(f)-f=P_{\theta}(f+g)-(f+g)$, hence

$$\int\left(P_{\theta^{\prime}}(f)-f\right)\left(\theta^{\prime}+dd^{c}P_{\theta^{\prime}}(f)\right)^{n}=\int\left(P_{\theta}(f+g)-(f+g)\right)\left(\theta+dd^{c}P_{\theta}(f+g)\right)^{n}$$

for all $f\in C^{0}(X)$. ∎