Lemma 5.28.35087175B2C39

Proof. The functions $m_\delta =\max (a+\delta ,b)$ satisfy $b\leq m_\delta \leq b+\delta $ and decrease to $b$ as $\delta \downarrow 0$. By lemma 5.24.1 and proposition 5.28.2, the products $\theta _{m_\delta }^ k\wedge W$ are positive, and by bounded decreasing continuity they converge weakly to $\theta _ b^ k\wedge W$. On the plurifine open set $\{ a+\delta >b\} $, which contains $\{ a=b\} $, we have $m_\delta =a+\delta $, so plurifine locality identifies the restriction of $\theta _{m_\delta }^ k\wedge W$ to this set with that of $\theta _ a^ k\wedge W$. Hence, for every compact set $K\subset \{ a=b\} $ in a coordinate chart, the upper semicontinuity of the mass of compact sets under weak convergence gives

Since the measures are locally finite, inner regularity extends this inequality to all Borel subsets of the contact set, and a countable cover by charts proves the assertion. $\square$