  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 \[ \theta_{P_{\theta}(f)}^n= { \bf
{1}}_{\{P_{\theta}(f) = f\}} \ \theta_f^n\qquad {\rm and }\qquad
\theta_{P_\theta[\varphi](f)}^n = { \bf {1}}_{\{P_\theta[\varphi](f) = f \}} \
\theta_f^n. \]
