Example 3.4.244B0199B64A0D

Example 3.4.24.4B0199B64A0D Let $d\geq 2$. The fundamental solution $E_ d$ is negative on $B(0,1)$ and equals $-\infty $ at the origin. The functions $u_ j=\max \{ -1,E_ d/j\} $, $j\geq 1$, are subharmonic on $B(0,1)$, take values in $[-1,0)$ and increase with $j$. Their limit is $0$ on $B(0,1)\setminus \{ 0\} $ and $-1$ at the origin, whereas its upper semicontinuous regularization is identically zero. Thus even an increasing sequence of bounded subharmonic functions can have a limit that is not upper semicontinuous.