Equilibrium potential of a finite set ($h_A$)

For a finite vertex set $A$, its equilibrium potential is $h_A(x)=P_x(H_A<\infty )$. It equals one on $A$, is harmonic off $A$, and belongs to the Dirichlet energy space with zero boundary at infinity.

The equilibrium measure is supported on $A$ and is given by

The capacity of a finite set is the total mass of its equilibrium measure of a finite set:Capacity is monotone under inclusion, though strict inclusion need not give strict inequality.

For a vertex $o$,