Proposition 18.6.1.47A414940A74 The formula
CE9BB18A9158
18.6.2
\begin{equation} d_{\mathcal I,\theta ,\mathrm{d}V}([u]_{\mathcal I},[v]_{\mathcal I}) =\int _ X|P_\theta [u]_{\mathcal I}-P_\theta [v]_{\mathcal I}|\, \mathrm{d}V \end{equation}
defines a finite metric on $\mathfrak S_{\mathcal I}(X,\theta )$. This metric space is totally bounded. Its metric completion is canonically the $L^1$ closure of $\mathfrak M_{\mathcal I}(X,\theta )$ inside the compact space $\mathcal K_\theta $ of normalized $\theta $-plurisubharmonic functions in theorem 5.25.2. Changing $\mathrm{d}V$ to another smooth positive probability volume form gives a bi-Lipschitz equivalent metric.
Proof.
The ideal envelopes are integrable and depend only on the ideal classes, so the formula is well defined and finite; symmetry and the triangle inequality are those of the $L^1$ norm. If the distance is zero, the two envelopes agree almost everywhere, hence everywhere by lemma 5.24.3, and projection shows that the two ideal classes coincide. Conversely, equal ideal classes have the same envelope.
By the normalized compactness theorem, $\mathfrak M_{\mathcal I}$ lies in the compact space $\mathcal K_\theta $. Every subset of a compact metric space is totally bounded, and its completion is its closure in that space. Finally, the ratio of two smooth positive volume forms on the compact manifold $X$ is bounded above and below by positive constants, and multiplying the integral by these bounds proves the last assertion.
$\square$
Pome
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