PomeLemma 11.37.1.2E4E82F8C3FC For every $\nu \in \mathcal P(X)$,
\[ \operatorname {Ent}_{\mathrm{d}V}(\nu ) =\sup _{F\in C(X)}\left(\int _ XF\, \mathrm{d}\nu -\log \int _ X\mathrm{e}^ F\, \mathrm{d}V\right). \]
11.37 Entropy and the closed case694B6AE5D5E5
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In a closed background, all potentials of an ordinary singularity type have the same mass, and integrating the equation determines the scalar explicitly; its convex dual is then the relative entropy. The variational formula for the entropy also gives estimates in nonclosed backgrounds.
For a Borel probability measure $\nu $ absolutely continuous with respect to $\mathrm{d}V$, let $h$ be its nonnegative density and define the relative entropy
\[ \operatorname {Ent}_{\mathrm{d}V}(\nu )=\int _ Xh\log h\, \mathrm{d}V, \qquad 0\log 0=0, \]
and set $\operatorname {Ent}_{\mathrm{d}V}(\nu )=+\infty $ if $\nu $ is not absolutely continuous. The integral is well defined in $(-\infty ,+\infty ]$, since $h\log h\geq -\mathrm{e}^{-1}$; it may be negative when $\mathrm{d}V(X)\ne 1$.
Proof. If $\nu $ is not absolutely continuous, choose a compact set $K$ with $\mathrm{d}V(K)=0$ and $\nu (K)>0$. For $t>0$, regularity gives a continuous cutoff $0\leq \chi _ t\leq 1$, equal to one on $K$, whose support has volume at most $\mathrm{e}^{-2t}$. For $F=t\chi _ t$, the expression in the supremum is at least
\[ t\nu (K)-\log \bigl (\mathrm{d}V(X)+\mathrm{e}^{-t}\bigr ), \]
which tends to infinity with $t$.
Suppose now that $\nu =h\, \mathrm{d}V$. If its entropy is finite, Jensen's inequality for the logarithm gives
\[ \int _ XF\, \mathrm{d}\nu -\operatorname {Ent}_{\mathrm{d}V}(\nu ) \leq \log \int _{\{ h>0\} }\mathrm{e}^ F\, \mathrm{d}V \leq \log \int _ X\mathrm{e}^ F\, \mathrm{d}V. \]
For infinite entropy this upper bound is automatic. For the opposite inequality, set
\[ F_ k=\log \bigl (\max \{ k^{-1},\min \{ h,k\} \} \bigr ),\qquad k\geq 1. \]
Then $\int _ X\mathrm{e}^{F_ k}\, \mathrm{d}V\to 1$ and $\int _ XF_ k\, \mathrm{d}\nu \to \operatorname {Ent}_{\mathrm{d}V}(\nu )$, with the value $+\infty $ allowed: the positive parts of the integrands $hF_ k$ increase to that of $h\log h$, and their negative parts are dominated by the integrable negative part of $h\log h$. For each fixed $k$, approximate $F_ k$ in $L^1(\mathrm{d}V+\nu )$ by continuous functions with the same bounds; both integrals, including the exponential one, converge to those of $F_ k$. Hence the supremum over continuous functions is at least the value of the expression at $F_ k$, and letting $k\to \infty $ completes the proof. $\square$
Suppose now that $\theta $ is a Kähler form. By the closed-background theory of Theorem 13.8.7, all potentials of the ordinary type of $\varphi $ have the same non-pluripolar mass $m_\varphi $, which is positive under the standing hypotheses. For $F\in L^\infty (\mathrm{d}V)$, integrating the equation gives
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11.37.2
\begin{equation} B_\varphi (F)=\log m_\varphi -\log \int _ X\mathrm{e}^ F\, \mathrm{d}V, \qquad \mathcal A_\varphi (F) =\left\{ \frac{\mathrm{e}^ F\, \mathrm{d}V}{\int _ X\mathrm{e}^ F\, \mathrm{d}V}\right\} . \end{equation}
For the second identity, Jensen's inequality shows, as in proposition 11.34.18, that the normalized exponential measure is a supporting probability; differentiating in bounded directions and testing the supporting inequality with both signs of the parameter shows that it is the only one.
Proposition 11.37.3.33DF3CE1122A If $\theta $ is Kähler, then
\[ H_\varphi (\nu )=\log m_\varphi +\operatorname {Ent}_{\mathrm{d}V}(\nu ). \]
Thus the minimum formula (11.35.3) becomes the usual variational formula for the logarithmic integral in (11.37.2).
Proof. Substitute (11.37.2) into the defining supremum and apply lemma 11.37.1. $\square$
The dual functional depends on the reference volume. If $\mathrm{d}V$ is replaced by $\mathrm{d}V'=\mathrm{e}^ a \mathrm{d}V$ for a continuous real $a$, the corresponding functionals satisfy
\[ B'_\varphi (F)=B_\varphi (F+a),\qquad H'_\varphi (\nu )=H_\varphi (\nu )-\int _ Xa\, \mathrm{d}\nu . \]
The first identity follows from the equation, and the second, which also holds when the values are infinite, from the change of variable $G\mapsto G+a$ in the defining supremum. In a nonclosed background, $H_\varphi $ is still defined by the same supremum, but the total mass need not be constant on an ordinary type, so the entropy computation no longer determines it.
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