PomeC5C10F2AB708Theorem 13.12.2.C5C10F2AB708 Let $\theta '$ be another closed real smooth $(1,1)$-form on $X$ representing a big cohomology class. Then there are continuous functions $f,g\colon [0,\infty )\rightarrow [0,\infty )$, with $f(0)=g(0)=0$, such that for any Borel subset $E\subseteq X$, we have
Proof. Use the comparison with Kähler capacity with the reference potentials defining $\operatorname{Cap}_\theta$ and $\operatorname{Cap}_{\theta'}$. It gives continuous nondecreasing functions $F,G,F',G'$, all zero at zero, with \[\operatorname{Cap}_\theta\leq F\circ\operatorname{Cap}_\omega,\quad\operatorname{Cap}_\omega\leq G\circ\operatorname{Cap}_\theta,\quad\operatorname{Cap}_{\theta'}\leq F'\circ\operatorname{Cap}_\omega,\quad\operatorname{Cap}_\omega\leq G'\circ\operatorname{Cap}_{\theta'}.\] The reference potentials have positive mass because the classes are big. Set $f=F\circ G'$ and $g=F'\circ G$. Composing the respective inequalities proves the two assertions for every Borel set. The compositions are continuous and vanish at zero, as required.
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