Theorem 5.13.3FDD1E63B4457

Proof. Suppose first that $\varphi\in L^1_{\mathrm{loc}}(\Omega)$ and $\Delta\varphi\geq0$. Apply the distributional characterization of subharmonic functions to the distribution defined by $\varphi$. It gives a subharmonic representative $\psi$ of this distribution. Equality of the distributions defined by two locally integrable functions is equality almost everywhere, so $\varphi=\psi$ almost everywhere.

Conversely, a subharmonic function on the connected domain $\Omega$ is locally integrable and has positive distributional Laplacian by the same theorem. Replacing it on a null set changes neither its distribution nor its local integrability. Finally, two subharmonic representatives that agree almost everywhere agree at every point: their values are the limits of their averages over shrinking balls. This also proves uniqueness. Multiplying the usual planar Laplacian by a fixed positive normalization constant does not change positivity.