Monge–Ampère measures on contact sets

Let $(X, \omega)$ be a compact Kahler manifold of complex dimension n and $\theta$ smooth closed real $(1,1)$-form on $X$ such that its cohomology class $\{ \theta \}\in H^{1,1}(X, \mathbb{R})$ is pseudoeffective. $\varphi$ $\theta$-psh function, let $f$ continuous function with bounded distributional laplacian respect to $\omega$ $\varphi \leq f. $ Then the non-pluripolar measure $\theta_\varphi^n:= (\theta + dd^c \varphi)^n$ satisfies equality: $$ {\bf{1}}_{\{ \varphi = f \}} \theta_\varphi^n \theta_f^n,$$ where, for subset $T\subseteq X$, ${\bf{1}}_T$ characteristic function. In particular we prove \[ \theta_{P_{\theta}(f)}^n= { \bf {1}}_{\{P_{\theta}(f) f\}} \theta_f^n\qquad {\rm }\qquad \theta_{P_\theta[\varphi](f)}^n {1}}_{\{P_\theta[\varphi](f) \theta_f^n. \]