Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 9 / 5 / a / Solution

For a real locally integrable function $f$, the distributional definition of a subharmonic function is $\Delta f\ge 0$: for every nonnegative $\phi \in C_c^{\infty }(\Omega )$,A superharmonic function has the opposite inequality, equivalently $-f$ is subharmonic. A harmonic function has $\Delta f=0$ distributionally, equivalently equality for every test function. These definitions concern almost-everywhere classes; the given upper semicontinuous representative supplies the pointwise version for subharmonic functions.