Pome

Corollary 18.7.7E941DFE18F1D

Corollary 18.7.7.E941DFE18F1D For the sequence above, the following are equivalent:

\[ P_\theta [u]=P_\theta [u]_{\mathcal I},\qquad q_ j\longrightarrow P_\theta [u]\text{ in }L^1, \qquad q_ j\searrow P_\theta [u]\text{ pointwise}. \]

They hold for one choice in (18.7.5) if and only if they hold for every such choice. For the fixed volume form $\mathrm{d}V$,

\[ \lim _ j\int _ X(q_ j-P_\theta [u])\, \mathrm{d}V =\int _ X(P_\theta [u]_{\mathcal I}-P_\theta [u])\, \mathrm{d}V\geq 0, \]

and the vanishing of this limit is a further equivalent condition.

Proof. The theorem identifies both limits of the $q_ j$ as $P_\theta [u]_{\mathcal I}$ for every such choice, and its $L^1$ convergence gives the integral formula. By proposition 18.4.4, the limiting difference is nonnegative. Its integral therefore vanishes exactly when the two envelopes agree almost everywhere, hence everywhere by lemma 5.24.3. This proves all the equivalences. $\square$

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