In this paper we have the following situation on page 60. $E$ is a compact subset of $\mathbb{R}^\tau\cup\{\infty\}$ (one point compactification) for $\tau\geq2$, $M_0$ is a point in the boundary of $E$ and $u$ is subharmonic on an open neighborhood $V$ of $M_0$. M. Brelot says that we can extend $u$ to a function that is subharmonic on a neighborhood of$E$, that he steel calls $u$, by "modifying properly $u$ outside $V$". Can someone explains the detail of this extension?

By the way a function $v$ on an open set $W$ of $\mathbb{R}^\tau\cup\{\infty\}$ is called subharmonic if: either $W$ does not contain $\infty$ and in this case $v$ is subharmonic on $W$ in the usual sense, or $W$ contains $\infty$ and then 1) $v$ is upper semicontinuous at $\infty$ and 2) $v(\infty)\leq$ the mean integral value of $u$ over any ball $B$ that is relatively compact in $W$.

Recall that the extension of a subharmonic function to $\mathbb{R}^\tau$ is classic. If $u$ is subharmonic on a neighborhood of $\overline{V}$ (closure), where $V$ is a bounded open set in $\mathbb{R}^\tau$ such that $\mathbb{R}^\tau\setminus\overline{V}$ is connected, then there exists a subharmonic function $\bar{u}$ on $\mathbb{R}^\tau$ that coincides with $u$ on $\overline{V}$ (see Armitage and Gardiner, "classical potential theory", pg192).

But Brelot extends $u$ to an open set of $\mathbb{R}^\tau\cup\{\infty\}$ containg $E$. How??

Sur le rôle du point à l'infini dans la théorie des fonctions harmoniques, Ann. Éc. Norm. sup., t. 61, 1944. I did not follow this reference, but I guess the argument is not much different from the classical one. $\endgroup$ – Mateusz Kwaśnicki Feb 26 at 8:13