We show that any continuous measure $\nu$ in the class of a generalized Gibbs stream on the boundary of a CAT( $-\kappa$ ) group $G$ arises as a harmonic measure for a random walk on $G$ . Under an additional mild hypothesis on $G$ and for $\nu$ , Hölder equivalent to a Gibbs measure, we show that $(\partial G,\nu)$ arises as a Poisson boundary for a random walk on $G$ . We also prove a new approximation theorem for general metric measure spaces giving quite flexible conditions for a set of functions to be a positive basis for the cone of positive continuous functions