Lemma 7.10 . [01H0]

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Lemma 7.10.

Let $u:X\to[-\infty,+\infty[$ be a function such that for each SNC model $\mathcal{X}$ we have

(i)

$u\circ i_{\mathcal{X}}$ is continuous on $\Delta_{\mathcal{X}}$ .
(ii)

$u\leq u\circ p_{\mathcal{X}}$ .

Then $u^{*}=\inf_{\mathcal{X}}u\circ p_{\mathcal{X}}$ , hence $u^{*}\circ i_{\mathcal{X}}=u\circ i_{\mathcal{X}}$ for all $\mathcal{X}$ .