Suppose $U$ is an open set in $\mathbb{R}^n$ ($n>1$), and $u: U\to (-\infty,\infty]$ locally bounded below. Suppose $u$ satisfies the superharmonic mean value inequality, i.e. for all ball $B(x,r)$ relatively compact in $U$, $u(x)$ is bounded below by the mean value of $u$ over the ball. In this case, this is a standard result that the lower semi-continuous regularization of $u$, i.e. $$u_*(x):=\liminf_{\substack{y\rightarrow x\\(u\in U)}}u(y),$$ is superharmonic on $U$.
I use this procedure and obtain incorrect results! Is there something else in the above result I mentioned that I am missing ?