Theorem 21.11.56`ED65AA5C2E92`

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. $\square$

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BibTeX citation

@misc{pome-tag-ED65AA5C2E92,
  author = {{Pome contributors}},
  title = {Theorem 21.11.56},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/ED65AA5C2E92/}
}

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Theorem 21.11.56ED65AA5C2E92

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Theorem 21.11.56`ED65AA5C2E92`