Lemma 5.43.10633794C1EFC2

Proof. Put $M_ j=\sup _ Cv_ j$ and $M=\sup _ Cv$. Hartogs' lemma applied at moving maximum points gives $\limsup M_ j\leq M$. A subsequence with limiting suprema below $M$ would give $v$ below that limit almost everywhere, hence everywhere, a contradiction. Thus $M_ j\to M$.

By surjectivity, $(v_ j-M_ j)\circ p$ has supremum zero. Normalized compactness, theorem 5.25.2, makes these pullbacks precompact in $L^1(X)$. On every relatively compact product chart over $C\setminus D$, where $D$ is the finite set of critical values, Fubini's theorem and bounded smooth volume densities give local $L^1$ convergence to $(v-M)\circ p$. Every subsequential limit therefore equals this pullback off $p^{-1}(D)$, and hence almost everywhere, because that analytic set has smooth volume zero. All subsequential limits agree, so the entire normalized sequence converges. Adding $M_ j\to M$ proves the assertion. $\square$