Let $f$ be a concave function on $N_{\mathbb{R}}$. The real Monge-Ampère measure of $f$ with respect to $\mu$ is defined, for a Borel subset $E$ of $N_{\mathbb{R}}$, as

It is a measure with support contained in ${\operatorname{dom}}(\partial f)$. The correspondence $f\mapsto{\mathcal{M}}_{\mu}(f)$ is called the Monge-Ampère operator.

When the measure $\mu$ is clear from the context, we will drop it from the notation. Moreover, since we are not going to consider complex Monge-Ampère measures, we will simply call ${\mathcal{M}}_{\mu}(f)$ the Monge-Ampère measure of $f$.