Positive linear combinations and finite maxima preserve subharmonicity

Let $I=\{j:{\alpha }_j>0\}$. If $I$ is empty, the combination is the harmonic zero function. Otherwise it means the well-defined extended-real sum ${\sum }_{j\in I}{\alpha }_j{u}_j$; no product $0\cdot (-\mathrm{\infty })$ occurs. Finite sums with positive coefficients preserve upper semicontinuity, and [L3] shows that all summands are finite simultaneously almost everywhere, so their sum is not identically $-\mathrm{\infty }$. On any closed disc, multiply the submean inequality for $u_j$ by ${\alpha }_j>0$ and sum over $I$ to obtain the submean inequality for the combination.

For the finite maximum, upper semicontinuity is preserved by finite maxima. Let $\overline{D(a,r)}\subseteq \mathrm{\Omega }$ and let $h$ be continuous on the closure, harmonic on the disc, and satisfy $h\ge \max (u_1,\dots ,u_m)$ on the boundary. Then $h\ge u_j$ on the boundary for every $j$, so [L1] gives $h\ge u_j$ throughout the disc for every $j$. Therefore $h\ge \max (u_1,\dots ,u_m)$ on the disc. Another use of [L1] shows that the maximum is subharmonic.

Steps 1.1 and 1.2 prove the two closure properties.