PomeProposition 21.11.42.A63267E9EA69 We have
\[ \mathcal{E}^{\infty }(N,P)\subseteq \mathcal{E}^{1}(N,P)\subseteq \mathcal{E}(N,P) \subseteq \mathcal{P}(N,P). \]
Proposition 21.11.42A63267E9EA69
Proof. When $\operatorname{Int}P=\varnothing $, the assertion is clear. We assume that $\operatorname{Int}P\neq \varnothing $. The second inclusion follows from the definition. We only handle the first inequality. Take $F\in \mathcal{E}^{\infty }(N,P)$. By definition, $F\sim \operatorname{Supp}_ P$ and hence $F^*\sim \chi _ P$. It follows that $P=\operatorname{Dom}F^*$.
By proposition 21.11.38, we already know that
\[ \nabla F(N)\subseteq P=\operatorname{Dom}F^*. \]
On the other hand, by proposition 21.11.34, we have
\[ \operatorname{Int}P\subseteq \nabla F(N). \]
So it follows that
\[ P=\overline{\nabla F(N)}. \]
It is clear that $F^*\sim \chi _ P$ is integrable. $\square$
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