Let \(B_ n\) be the unit ball in \(\mathbb{C}^ n\), \(n \geq 1\), \(S=\partial B_ n\). An inner function \(f\) on \(B_ n\) is said to be good if \[ \lim_{r \to 1} \int_ S \ln | f(rt) | d \sigma(t)=0, \] where \(\sigma\) is Haar's measure on \(S\). Let \(G_ \mu\) be the Green potential of regular Borel measure \(\mu\) on \(B_ 1\) satisfying \((1-| w |) \in {\mathcal L}^ 1 (\mu)\). The author characterizes such a measure \(\mu\) such that \(G_ \mu \circ f\) is a good plurisuperharmonic function on \(B_ n\) (or polydisk \(U^ n)\) for every inner function \(f\) on \(B_ n\) (or \(U^ n)\) (Theorem 1). Then this result is used to obtain some facts of the boundary behavior of \(G_ \mu(f(rt))\). Let us formulate one of them: Theorem 6. Let \(f\) be an inner function on \(B_ n\). Then \(\liminf_{r \to 1}(1-r)G_ \mu(f(rt))=0\) for all \(t \in \partial B_ n\) for which \(\lim_{r \to 1} | f(rt) |=1\).