Monge-Ampère measures on contact sets

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