PomeTheorem 14.19.2 (Kołodziej–Nguyen).C8D798C2D2F0 There are $c>0$ and $u\in \mathrm{PSH}(X,\omega )\cap C(X)$ such that
\[ (\omega +\mathrm{dd}^{\mathrm{c}}u)^ n=c f\, \mathrm{d}V. \]
The potential can be normalized by $\sup _ Xu=0$.
Theorem 14.19.2 (Kołodziej–Nguyen)C8D798C2D2F0
of theorem 14.19.2. Choose smooth strictly positive functions $f_ j\to f$ in $L^ p(X,\mathrm{d}V)$. By theorem 14.2.1, there are smooth $u_ j$ with $\omega _{u_ j}>0$ and constants $c_ j$ such that
\[ \omega _{u_ j}^ n=c_ jf_ j\, \mathrm{d}V,\qquad \sup _ Xu_ j=0,\qquad c_ j>0. \]
We first bound the constants. For $n\geq 2$, fix a Gauduchon metric $\gamma $. By proposition 11.5.4,
\[ c_ j^{1/n}A_ j\leq \int _ X\omega \wedge \gamma ^{n-1}, \qquad A_ j=\int _ X(f_ j\mathrm{d}V/\gamma ^ n)^{1/n}\gamma ^ n. \]
Since $|x^{1/n}-y^{1/n}|\leq |x-y|^{1/n}$ for $x,y\geq 0$, Hölder's inequality gives $A_ j\to A>0$, where $A=\int _ X(f\mathrm{d}V/\gamma ^ n)^{1/n}\gamma ^ n$. Thus $c_ j$ is bounded above. In dimension one, Stokes' theorem gives the same estimate, with $A_ j=\int _ Xf_ j\, \mathrm{d}V$.
The lower bound for the density norm in theorem 14.19.19 gives $c_ j\| f_ j\| _ p\geq c_ p$, so the constants are bounded away from zero. The densities $c_ jf_ j$ are therefore uniformly bounded in $L^ p$, and the same theorem gives a common bound for the $u_ j$. By theorem 5.25.2, we may pass to a subsequence with $c_ j\to c>0$ and $u_ j$ Cauchy in $L^1$. By theorem 14.19.21, $u_ j\to u$ uniformly, with $u$ continuous and $\sup _ Xu=0$.
Bounded Bedford–Taylor continuity gives $\omega _{u_ j}^ n\rightharpoonup \omega _ u^ n$, while $c_ jf_ j\to cf$ in $L^ p$. Hence $\omega _ u^ n=cf\, \mathrm{d}V$, as required. $\square$
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