Atomic approximation on finite-dimensional spaces of continuous functions ($\left|\int fd\mu -{\sum }_i{t}_{i}f(w_i)\right|\le \epsilon \Vert f\Vert$)

A finite regular complex measure of total variation one can be approximated on a finite-dimensional vector subspace of continuous functions by a finite sum of phased point evaluations with ${\sum }_i|t_i|=1$. The Krein-Milman theorem and the extreme points of the dual unit ball of C(K) give weak-star approximation by convex combinations of phased point masses. A finite norm net of the vector subspace unit ball turns finitely many scalar approximations into one uniform estimate. Repeated nodes may be kept separate to retain the exact coefficient-magnitude sum.