# Continuity of the real Monge Ampère operator on convex functions

Let $E\subset\mathbb{R}^n$ be a convex set, $u:E\to\mathbb{R}$ a convex function and $B\subset E$ a Borel set.

We define the (multivalued) gradient \begin{array}{rccl} \nabla [u]:& \mathbb{R}^n & \to & \mathcal{P}((\mathbb{R}^n)^\vee)\\ & {a} & \mapsto & \{ \xi\in(\mathbb{R}^n)^\vee \mid {x}\mapsto \langle \xi, {x}\rangle -u({x}) \text{ attains its absolute maximum at }{a}\}. \end{array}

Informally, $\nabla [u]({a})$ is the set of slopes of all hyperplanes touching the graph at the point $({a},u({a}))\in \mathbb{R}^n\times\mathbb{R}$ of the function and staying below.

Also we define $\nabla[u](B):=\bigcup_{{x}\in B} \nabla[u]({x})$.

Finally the Monge-Ampère measure of $u$ of $B$ is $M[u](B):=\lambda_L(\nabla[u](B))$, where $\lambda_L$ is the Lebesgue measure. In some sense it measures the total convexity of $u$ in $B$.

Is it easy from this definition to see that $M:${convex functions}$\to${$\mathbb{R}^n$positive measures} is continuous for the locally uniform convergence topology and the weak topology, respectively?

(I am reading the article of Passare and Rullgard "Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope")

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