Say we define subharmonic functions as functions $f(x) \in C^0(\Omega)$ such that for any interior $x \in \Omega, \exists \rho>0$ such that for any $0<r<\rho$, $$ f(x) \leq \frac{1}{|B_r(x)|}\int_{B_r(x)}f(y)dy. $$
How would we go about deducing the global mean value inequality, i.e.
$$ f(x) \leq \frac{1}{|B_r(x)|}\int_{B_r(x)}f(y)dy, \forall B_r(x) \subset \Omega?$$
Here, $|B_r(x)|$ denotes the volume of $B_r(x)$.
Thanks