= Locally integrable supremum theorem for subharmonic functions
{title2=$g=\sup_j f_j\in L^1_{\mathrm{loc}}\Longrightarrow\Delta g\ge0$}
For a countable family of <subharmonic functions>, a <locally integrable> pointwise supremum is subharmonic in the <distributional derivative> sense. Finite maxima preserve subharmonicity: approximate the maximum by the smooth convex coordinatewise increasing function $(s+t+\sqrt{(s-t)^2+\delta^2})/2$, apply the <chain rule> after mollification, then pass to a local <L1 norm> convergence limit. The maxima $g_N=\max(f_1,\ldots,f_N)$ increase to $g$ and are bounded in absolute value by $|f_1|+|g|$ on compact sets, so the <dominated convergence theorem> completes the proof. The hypothesis cannot be omitted: $f_j=j$ has infinite supremum. To obtain a pointwise <subharmonic function>, take the <canonical representative of a subharmonic distribution>.
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