Let $\Omega\subseteq\mathbb{R}^{n}$ be bounded, open and convex and denote by $\lambda$ the standard Lebesgue measure on $\mathbb{R}^{n}$ and by $\langle\cdot,\cdot\rangle$ the standard scalar product on $\mathbb{R}^{n}$. Let $h$ be a convex function on $\Omega$ and $x_{0}\in\Omega$. We define the gradient image of $x_{0}$ under $h$ to be

$$\nabla h(x_{0}):=\left\{p\in\mathbb{R}^{n}\;\Big|\;\forall x\in\Omega\;:\;h(x_{0})+\langle x-x_{0},p\rangle\leq h(x)\right\}$$

and for $E\subseteq\Omega$

$$\nabla h(E):=\bigcup_{x_{0}\in E}\nabla h(x_{0}).$$

Note that if $E$ is a Borel set, the same is true for $\nabla h(E)$. Finally we define the Monge-Ampère measure associated to $h$ by

$$\MA(h)(E):=\lambda(\nabla h(E))$$

for all Borel sets $E\subseteq\Omega$. It is indeed a measure on the Borel $\sigma$-algebra, for details see [RT77, Section 2]. The real Monge-Ampère operator is continuous in the sense that if $(u_{n})_{n\in\mathbb{N}}$ is a sequence of convex functions on $\Omega$ converging pointwise to a convex function $u$ then $(\MA(u_{n}))_{n\in\mathbb{N}}$ converges weakly to $\MA(u)$. If $h$ is two times continuously differentiable then $\MA(h)=\det D^{2}h\cdot\lambda$.