= Canonical representative of a subharmonic distribution
{title2=$\widetilde u(x)=\lim_{r\downarrow0}\frac{1}{|B_r(x)|}\int_{B_r(x)}u(y)\,dy$}
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.
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