Weighted pluripotential theory in CN

Let K ⊂ C^N be compact and let w be a nonnegative, uppersemicontinuous function on K with {z ∈ K: w(z) > 0} nonpluripolar. Let Q := - log w and define the weighted pluricomplex Green function V*_K,Q(z) = lim sup_ζ→zV_K,Q(ζ) where V_K,Q(z) := sup{u(z): u plurisubharmonic in C^N, u(z) ≤ log⁺ |z| + C, u ≤ Q on K} (C depends on u). If w ≡ 1; i.e., Q ≡ 0, we are in the unweighted case and we write V_K := V_K,0. We prove weighted generalizations of several results in pluripotential theory, and we prove a version of a unique continuation property of maximal psh functions. These results are used to show that if E ⊂ F are compact subsets of C^N, then d(E) = d(F) if and only if V*_E = V*_F where d(E) denotes the transfinite diameter of E.