PomeC745DD03819CProposition 16.27.8.C745DD03819C Suppose that $\{ \theta \} $ is big, and let $\phi $ be a model potential of mass $m>0$. There are $\alpha ,C>0$, depending only on $X$, $\theta $ and $\omega $, such that every Borel set $E\subset X$ satisfies
7D19131D8004
16.27.9
where the right side is interpreted as zero when the capacity is zero.
Proof. Let $\alpha $ and $C_0$ be the constants of corollary 10.3.3. First let $K$ be compact and non-pluripolar, and let $S,L$ be as in lemma 16.27.5. The potential $p=S-L$ has supremum zero and satisfies $p\leq -L$ quasi-everywhere on $K$, because $S=\phi \leq 0$ there. Since pluripolar sets have zero volume, (10.3.4) gives
Write $c=\operatorname {Cap}_{\theta ,\phi }(K)$, which is positive by (16.27.6). By the same inequality,
so (16.27.9) holds for $K$ with $C=C_0\mathrm e^\alpha $. If $K$ is pluripolar, its volume and its capacity both vanish.
Now let $E$ be a Borel set. If $\operatorname {Cap}_{\theta ,\phi }(E)>0$, the estimate for compact subsets $K\subset E$, together with the monotonicity of capacity and the inner regularity of the smooth volume, gives (16.27.9). If $\operatorname {Cap}_{\theta ,\phi }(E)=0$, every compact subset of $E$ has zero capacity, hence is pluripolar by (16.27.6) and has zero volume; by inner regularity, $E$ has zero volume. $\square$
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