Canonical representative of a subharmonic distribution ($\tilde{u}(x)={\lim }_{r\downarrow 0}\frac{1}{|B_r(x)|}{\int }_{B_r(x)}u(y)dy$)

For locally integrable real $u$ with nonnegative distributional derivative $\Delta u$, its ball means increase with radius. To see this, mollify: for a smooth subharmonic function the derivative of its spherical mean is the integral of $\Delta u$ over the ball divided by the sphere area. The ball means inherit monotonicity; local $L^1$ convergence passes this to $u$. Their decreasing-radius limit exists, may be minus infinity at exceptional points, and equals $u$ almost everywhere by the Lebesgue differentiation theorem. It is upper semicontinuous as an infimum of continuous ball means and satisfies the mean-value inequality. Arbitrary changes of a representative on a null set can destroy its pointwise maximum principle.