Behnke-Stein: increasing unions of pseudoconvex domains

If $\mathrm{\Omega }={\mathbb{C}}^n$, then [F2] is exactly the conclusion, so assume from now on that $\mathrm{\Omega }\mathrm{\ne }{\mathbb{C}}^n$; then $F:={\mathbb{C}}^n\setminus \mathrm{\Omega }$ is nonempty, and writing ${\delta }_j:={\delta }_{{\mathrm{\Omega }}_j}$ and $\delta :={\delta }_{\mathrm{\Omega }}$ in the sense of [F3] one has $0<{\delta }_j(a)<+\mathrm{\infty }$ for every $a\in {\mathrm{\Omega }}_j$ (positivity because ${\mathrm{\Omega }}_j$ is open, finiteness because $F\subseteq {\mathbb{C}}^n\setminus {\mathrm{\Omega }}_j\mathrm{\ne }\mathrm{\varnothing }$) and $0<\delta (a)<+\mathrm{\infty }$ for every $a\in \mathrm{\Omega }$.

For $a\in {\mathrm{\Omega }}_j$ the inclusion ${\mathrm{\Omega }}_j\subseteq {\mathrm{\Omega }}_{j+1}\subseteq \mathrm{\Omega }$ implies ${\mathrm{\Delta }}_r(a)\subseteq {\mathrm{\Omega }}_j\Rightarrow {\mathrm{\Delta }}_r(a)\subseteq {\mathrm{\Omega }}_{j+1}\Rightarrow {\mathrm{\Delta }}_r(a)\subseteq \mathrm{\Omega }$ for every $r>0$, hence ${\delta }_j(a)\le {\delta }_{j+1}(a)\le \delta (a)$ and, if $a\in {\mathrm{\Omega }}_{J_0}$, the eventual-tail limit $\gamma (a):={\lim }_{j\to \mathrm{\infty },j\ge J_0}{\delta }_j(a)={\sup }_{j\ge J_0}{\delta }_j(a)$ exists and is independent of $J_0$; moreover $\gamma (a)=\delta (a)$, because for every $r$ with $0<r<\delta (a)$ and every $r^{\mathrm{\prime }}$ with $r<r^{\mathrm{\prime }}<\delta (a)$ the definition [F3] gives ${\mathrm{\Delta }}_{r^{\mathrm{\prime }}}(a)\subseteq \mathrm{\Omega }$, so the closed polydisc ${\bar{\mathrm{\Delta }}}_r(a)\subseteq {\mathrm{\Delta }}_{r^{\mathrm{\prime }}}(a)\subseteq \mathrm{\Omega }$ of [F4] is closed and bounded in ${\mathbb{C}}^n$, hence compact by [F5] read through [F6], and is therefore covered by finitely many members of the increasing open cover $({\mathrm{\Omega }}_j{)}_{j\ge 1}$ of $\mathrm{\Omega }$, whose largest index, increased to $J\ge J_0$ if necessary, satisfies ${\bar{\mathrm{\Delta }}}_r(a)\subseteq {\mathrm{\Omega }}_J$ and hence $\gamma (a)\ge {\delta }_J(a)\ge r$; letting $r\uparrow \delta (a)$ gives $\gamma (a)=\delta (a)$.

Let $K\subseteq \mathrm{\Omega }$ be compact and choose $J$ with $K\subseteq {\mathrm{\Omega }}_J$ (the same finite-subcover argument applied to the increasing cover $({\mathrm{\Omega }}_j)$ of $K$); then for every $j\ge J$ the function $-\log {\delta }_j$ is plurisubharmonic on ${\mathrm{\Omega }}_j$ by the hypothesis that ${\mathrm{\Omega }}_j$ is Hartogs pseudoconvex and [F1], hence on the smaller domain ${\mathrm{\Omega }}_J$, the sequence $(-\log {\delta }_j{)}_{j\ge J}$ is decreasing on ${\mathrm{\Omega }}_J$ by step 2.1, and it converges pointwise on ${\mathrm{\Omega }}_J$ to $-\log \delta$ by the identity $\gamma =\delta$ of step 2.1; the limit is real-valued on ${\mathrm{\Omega }}_J$ because $0<\delta <+\mathrm{\infty }$ there by step 1.1, so it is not identically $-\mathrm{\infty }$ on any component and [F7] makes $-\log \delta$ plurisubharmonic on ${\mathrm{\Omega }}_J$.

Every point $a\in \mathrm{\Omega }$ lies in some ${\mathrm{\Omega }}_J$, an open neighbourhood of $a$ on which $-\log \delta$ is plurisubharmonic by step 3.1 applied with $K=\{a\}$; plurisubharmonicity is a local condition, so $-\log \delta$ is plurisubharmonic on $\mathrm{\Omega }$ and [F1] makes $\mathrm{\Omega }$ Hartogs pseudoconvex; together with the whole-space case of step 1.1 this proves the statement in both cases.