Definition 21.11.50FF5885207F9E

Definition 21.11.50.FF5885207F9E Let $F$ be a finite convex function on $N_{\mathbb {R}}$. We define the real Monge–Ampère measure $\operatorname{MA}_{\mathbb {R}}F$ as the Alexandrov measure on $N_{\mathbb {R}}$ given as follows: for each Borel measurable set $E\subseteq N_{\mathbb {R}}$, define

The measurability and countable additivity in this definition are standard properties of the subgradient map; see for example [Section 2.1, Fig17].