If the measure of $D$ is zero then both sides of equation (3.98) are zero. Therefore, the theorem is trivially true in this case. Thus, we may assume that $D$ has non-empty interior. Since $\operatorname{stab}(f)$ is compact, the right-hand side of (3.98) is continuous with respect to uniform convergence of functions, thanks to Proposition 3.18. Moreover, Proposition 3.93 and the fact that ${\mathcal{M}}_{\mu}(f)$ is finite imply that the left-hand side is also continuous with respect to uniform convergence. By the compacity of $D$, we can find a sequence of strictly concave smooth functions $(f_{n})_{n\geq 1}$ that converges uniformly to $f$. Hence, we may assume that $f$ is smooth and strictly concave. In this case, the Legendre transform $\nabla f\colon N_{\mathbb{R}}\to D^{\circ}$ is a diffeomeorphism.

By the definition of the Monge-Ampère measure,

(3.99)

$$-n!\int_{N_{\mathbb{R}}}f{\mathcal{M}}_{\mu}(f)=-n!\int_{D}f((\nabla f)^{-1}x)\,\text{\rm d}\mu(x),$$

which, in particular, shows that the integral on the left is convergent for smooth strictly concave functions with compact stability set. Therefore, it is convergent for any concave function within the hypothesis of the theorem.

By the properties of the Legendre transform,

(3.100)

$$-f((\nabla f)^{-1}(x))=f^{\vee}(x)-\langle(\nabla f)^{-1}(x),x\rangle.$$

Moreover,

	$\displaystyle\,\text{\rm d}(f^{\vee}\lambda)(x)$	$\displaystyle=\,\text{\rm d}f^{\vee}\land\lambda(x)+f^{\vee}\,\text{\rm d}\lambda(x)$
		$\displaystyle=\langle\nabla f^{\vee}(x),x\rangle\omega+nf^{\vee}\omega$
(3.101)			$\displaystyle=\langle(\nabla f)^{-1}(x),x\rangle\omega+nf^{\vee}\omega$

The result is obtained by combining equations (3.99), (3.100) and (3.101) with Stokes’ theorem. ∎