It is well known that amenability of a discrete (quantum) group is equivalent to the existence of a net of finitely supported (quantum) positive-definite functions converging pointwise to 1. We will show that using the (quantum) Godement mean one can weaken the latter condition to the existence of a net of finitely supported normalised (quantum) positive-definite functions which is `pointwise strictly positive in the limit'. This further implies that von Neumann algebras of unimodular discrete quantum groups enjoy a strong form of non-w*-CPAP, which we call the matrix epsilon-separation property. Based on the joint work with Jacek Krajczok.