PomeLemma 10.28.6.66D32D3D118D Let $u$ be a plurisubharmonic function on a product of disks $D_\zeta \times D_ t$ with $0\in D_ t$, and suppose that $u$ is not identically $-\infty $. Suppose that $u(\cdot ,t)$ is harmonic for almost every $t$. Then $u(\cdot ,0)$ is harmonic or identically $-\infty $.
Lemma 10.28.666D32D3D118D
Proof. Since $u$ is locally integrable, Fubini's theorem gives $\partial _\zeta \partial _{\bar\zeta }u=0$ in the sense of distributions. On smaller product disks, form the averages
\[ u_\varepsilon (\zeta )=\frac1{\pi \varepsilon ^2} \int _{|t|<\varepsilon }u(\zeta ,t)\, \mathrm{d}A(t), \]
where $\mathrm{d}A$ is planar area. Local upper bounds for $u$ and the reverse Fatou lemma give upper semicontinuity, and Fubini's theorem gives the submean inequality in $\zeta $. The averages are locally integrable, since $u$ is integrable on compact product disks. Thus each $u_\varepsilon $ is subharmonic with zero distributional Laplacian, hence harmonic.
Since $u$ is subharmonic in $t$, the disk averages decrease to $u(\zeta ,0)$ as $\varepsilon \downarrow 0$. A decreasing limit of harmonic functions is harmonic or identically $-\infty $: if the values are bounded below at one point, Harnack's inequality, applied to the differences from the first function, bounds the differences on compact subdisks, and compactness of harmonic functions gives a harmonic limit. This proves the alternative on every smaller disk, and hence on $D_\zeta $. $\square$
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