Studying the mean spherical mean and the volumetric mean, this question has occurred to me.
The volumetric mean is defined as follows:
Let $\Omega \subset \mathbb{R}^n$ open and $f:\Omega \to \mathbb{R}$ is harmonical.
If $\overline{B(x_0,r)} \subset \Omega$ then \begin{equation} f(x_0)=\frac{1}{|B(x_0,r)|}\int\limits_{B(x_0,r)}f(x)dx \end{equation}
My question is: can we extend $f$ such that we only need $\overline{B(x_0,r)} \subset \overline{\Omega}$ to use the volumetric mean?
This occurred to me when I realized $\overline{B(x_0,r)} \not\subset \Omega$, which violates the condition for the volumetric mean, shown above.