Let $N$ be a free Abelian group of finite rank $n$ (i.e., a lattice) and $M$ be its dual lattice. There is a canonical Lebesgue-type measure on $M_{\mathbb {R}}$, denoted by $\mathrm{d}\operatorname{vol}$, normalized so that the smallest cubes in $M$ have volume $1$. Similarly, the canonical measure on $N_{\mathbb {R}}$ is normalized in the same way and is denoted by $\mathrm{d}\operatorname{vol}$ as well.
We will write
\[ N_{\mathbb {R}}=N\otimes _{\mathbb {Z}}\mathbb {R},\quad M_{\mathbb {R}}=M\otimes _{\mathbb {Z}}\mathbb {R}. \]
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
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21.11.51
\begin{equation} \operatorname{MA}_{\mathbb {R}}F(E)\coloneqq n!\int _{\nabla F(E)} \mathrm{d}\operatorname{vol}. \end{equation}
The measurability and countable additivity in this definition are standard properties of the subgradient map; see for example [Section 2.1, Fig17].
Proposition 21.11.52.84CD4F8DE81D Suppose that $F\in C^{1,1}(N_{\mathbb {R}})\cap \mathrm{Conv}(N_{\mathbb {R}})$. Fix an identification $N=\mathbb {Z}^ n$. Then
\[ \operatorname{MA}_{\mathbb {R}}F=n!\cdot \det \nabla ^2 F \, \mathrm{d}\operatorname{vol}. \]
Proof. The measure normalization here is $n!$ times the Lebesgue measure of the subgradient image. The density formula for locally $C^{1,1}$ convex functions proves that this measure is $n!\det D^2F\,\mathrm dx$. Its hypotheses hold because $C^{1,1}$ implies that the gradient is locally Lipschitz, and the lattice coordinates identify $\mathrm dx$ with the normalized $\mathrm d\operatorname{vol}$. The Hessian is its almost-everywhere derivative, equivalently the density of the distributional Hessian. Thus the formula includes nonsmooth $C^{1,1}$ functions and has the required factor $n!$.
See [Example 2.2, Fig17].
Proposition 21.11.53.F30A95FDA78E Let $P\subseteq M_{\mathbb {R}}$ be a convex body and $F\in \mathcal{P}(N_{\mathbb {R}},P)$. Then $F\in \mathcal{E}(N_{\mathbb {R}},P)$ if and only if
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\begin{equation} \int _{N_{\mathbb {R}}}\operatorname{MA}_{\mathbb {R}}F=n! \operatorname{vol}P. \end{equation}
Proof.
Since $F\in \mathcal{P}(N_{\mathbb {R}},P)$, the function $F$ is closed and has domain $N_{\mathbb {R}}$. By proposition 21.11.34, $\overline{\nabla F(N_{\mathbb {R}})}=\overline{\operatorname{Dom}F^*}$, so this closure is convex. As $\nabla F(N_{\mathbb {R}})$ contains the relative interior of $\operatorname{Dom}F^*$, the definition of $\operatorname{MA}_{\mathbb {R}}$ shows that (21.11.54) is equivalent to
\[ \operatorname{vol}\overline{\nabla F(N_{\mathbb {R}})}=\operatorname{vol}P. \]
We first handle the case where $\operatorname{Int}P\neq \varnothing $. By proposition 21.11.38, $\overline{\nabla F(N_{\mathbb {R}})}\subseteq P$. Hence the latter is equivalent to
\[ \overline{\nabla F(N_{\mathbb {R}})}=P. \]
Indeed, a proper closed subset of the full-dimensional convex body $P$ has complement containing a non-empty relatively open subset of $P$, hence has strictly smaller volume.
Now assume that $\operatorname{Int}P=\varnothing $, then $\operatorname{vol}\overline{\nabla F(N_{\mathbb {R}})}=\operatorname{vol}P=0$ by proposition 21.11.38. The assertion is clear.
$\square$
Theorem 21.11.55.2DC9BC6F295A Let $F,F_ j\in \mathcal{P}(N_{\mathbb {R}},P)$ ($j\in \mathbb {Z}_{>0}$). Assume that $F_ j\to F$. Then
\[ \operatorname{MA}_{\mathbb {R}}(F_ j)\rightharpoonup \operatorname{MA}_{\mathbb {R}}(F) \]
weakly.
Qualification and proof. The topology on $\mathcal P(N_{\mathbb R},P)$ is pointwise convergence. Since all these functions are finite and convex, local uniform convergence of convex functions shows that $F_j\to F$ uniformly on compact subsets. Alexandrov-measure continuity now gives convergence against every compactly supported continuous test function, that is, weak convergence as Radon measures on this noncompact space.
Convergence against all bounded continuous functions would require an additional no-loss-of-mass condition and is not automatic here. For example, on $\mathbb R$ the functions $F_j(x)=\log(1+\exp(x-j))$ have common growth body $[0,1]$ and converge locally uniformly to $0$, while their real Monge--Ampère measures have total mass $1$ and move to $+\infty$. Their vague limit is zero. Thus the weak topology in this assertion must be read as the compactly supported test-function topology; if total masses also converge, a compact-cutoff argument upgrades it to the bounded-continuous topology.
See [Proposition 2.6, Fig17].
There is a well-known comparison principle.
Theorem 21.11.56.ED65AA5C2E92 Let $F,F'\in \mathcal{P}(N_{\mathbb {R}},P)$. Assume that $F\preceq F'$. Then
\[ \overline{\nabla F(N_{\mathbb {R}})}\subseteq \overline{\nabla F'(N_{\mathbb {R}})},\quad \int _{N_{\mathbb {R}}} \operatorname{MA}_{\mathbb {R}}(F)\leq \int _{N_{\mathbb {R}}} \operatorname{MA}_{\mathbb {R}}(F'). \]
Proof.
It suffices to observe that $F'{}^*\preceq F^*$, and hence the first assertion follows from proposition 21.11.34. By the definition of the real Monge–Ampère measure,
\[ \int _{N_{\mathbb R}}\operatorname{MA}_{\mathbb R}F=n!\operatorname{vol}\overline{\nabla F(N_{\mathbb R})}. \]
So the second assertion follows from the first.
$\square$
Lemma 21.11.57.8DE662F4EFE6 Let $F\in \mathcal{E}(N_{\mathbb {R}},P)$ and set $G=F^*$. Then the map $\nabla G$ pushes forward $n!\, \mathrm{d}\operatorname{vol}|_{P}$ to $\operatorname{MA}_{\mathbb R}(F)$. Equivalently, for every bounded Borel function $h$ on $N_{\mathbb R}$,
\[ \int _{N_{\mathbb R}} h\, \operatorname{MA}_{\mathbb R}(F)= n!\int _ P h(\nabla G(m))\, \mathrm{d}\operatorname{vol}(m). \]
Proof.
If $\operatorname{vol}P=0$, the assertion is clear from proposition 21.11.38 and the definition of $\operatorname{MA}_{\mathbb R}$. We may therefore assume that $P$ has non-empty interior.
Since $F\in \mathcal P(N_{\mathbb R},P)$, we have $\operatorname{Dom}G\subseteq P$ by proposition 21.11.38. On the other hand, by the definition of $\mathcal E(N_{\mathbb R},P)$ and proposition 21.11.34,
\[ P=\overline{\nabla F(N_{\mathbb R})}\subseteq \overline{\operatorname{Dom}G}. \]
Hence $\overline{\operatorname{Dom}G}=P$. Since $\operatorname{Dom}G$ is convex, $G$ is finite on $\operatorname{Int}P$ and is equal to $\infty $ outside $P$. Let $D$ be the set of points in $\operatorname{Int}P$ at which $G$ is differentiable. By [Theorem 25.5, Roc70], $P\setminus D$ has measure zero. By [Theorem 23.5, Roc70], $m\in \nabla F(n)$ if and only if $n\in \nabla G(m)$. Hence for any Borel set $E\subseteq N_{\mathbb R}$,
\[ \{ m\in D:\nabla G(m)\in E\} =D\cap \nabla F(E). \]
It follows from definition 21.11.50 that
\[ n!\operatorname{vol}\{ m\in P:\nabla G(m)\in E\} =n!\operatorname{vol}\nabla F(E)=\operatorname{MA}_{\mathbb R}(F)(E). \]
Our assertions follow.
$\square$
Finally, let us recall a mixed version of the real Monge–Ampère measure:
Theorem 21.11.58.59E2455286B4 There is a symmetric map assigning to every $n$-tuple of finite convex functions $F_1,\ldots ,F_ n$ on $N_{\mathbb R}$ a positive Radon measure
\[ \operatorname{MA}_{\mathbb R}(F_1,\ldots ,F_ n) \]
on $N_{\mathbb R}$ with the following properties:
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For every $q\in \mathbb Z_{>0}$, finite convex functions $F_1,\ldots ,F_ q$ on $N_{\mathbb R}$, and $\lambda _1,\ldots ,\lambda _ q\geq 0$,
\[ \operatorname{MA}_{\mathbb R}(\lambda _1F_1+\cdots +\lambda _ qF_ q) =\sum _{i_1,\ldots ,i_ n=1}^ q \lambda _{i_1}\cdots \lambda _{i_ n} \operatorname{MA}_{\mathbb R}(F_{i_1},\ldots ,F_{i_ n}). \]
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For every finite convex function $F$ on $N_{\mathbb R}$,
\[ \operatorname{MA}_{\mathbb R}(F,\ldots ,F)=\operatorname{MA}_{\mathbb R}(F). \]
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The map $\operatorname{MA}_{\mathbb R}$ is additive and positively homogeneous of degree $1$ in each entry. In other words, for finite convex functions $F,G,F_2,\ldots ,F_ n$ on $N_{\mathbb R}$ and $\lambda ,\mu \geq 0$,
\[ \operatorname{MA}_{\mathbb R}(\lambda F+\mu G,F_2,\ldots ,F_ n) =\lambda \operatorname{MA}_{\mathbb R}(F,F_2,\ldots ,F_ n) +\mu \operatorname{MA}_{\mathbb R}(G,F_2,\ldots ,F_ n). \]
Proof. The construction of mixed Alexandrov measures gives a positive, symmetric, additive and positively homogeneous measure in the $n$ arguments, with diagonal equal to the Alexandrov measure defined here. Repeatedly expand the diagonal measure of $\sum_i\lambda_iF_i$ using this additivity and homogeneity. Each ordered choice $(i_1,\ldots,i_n)$ contributes $\lambda_{i_1}\cdots\lambda_{i_n}\operatorname{MA}_{\mathbb R}(F_{i_1},\ldots,F_{i_n})$, which is the asserted polynomial formula.
The earlier proof first verifies positivity for smooth convex functions using mixed determinants, and then uses convolution and local uniform continuity of the diagonal measure to pass to finite convex functions. In particular positivity is proved rather than inferred from a signed polarization formula. Uniqueness follows by recovering each coefficient from the diagonal polynomial, for instance by polarization. The normalization remains $n!$ on the diagonal.
This is [Theorem 4.3(a),(b),(f), CLM22], multiplied by $n!$ to match the normalization in definition 21.11.50. The positivity is part of the construction: it is the non-negativity of the mixed discriminant in the smooth case and follows in general by weak continuity. The mixed real Monge–Ampère measure was also studied in [TW02, PR04].
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