Let $μ$ be a Borel probability measure on a compact path-connected metric space $(X, ρ)$ for which there exist constants $c,β>1$ such that $μ(B) \geq c r^β$ for every open ball $B\subset X$ of radius $r>0$. For a class of Lipschitz functions $Φ:[0,\infty)\to R$ that piecewisely lie in a finite-dimensional subspace of continuous functions, we prove under certain mild conditions on the metric $ρ$ and the measure $μ$ that for each positive integer $N\geq 2$, and each $g\in L^\infty(X, dμ)$ with $\|g\|_\infty=1$, there exist points $y_1, \ldots, y_{ N}\in X$ and real numbers $λ_1, \ldots, λ_{ N}$ such that for any $x\in X$, \begin{align*} & \left| \int_X Φ(ρ(x, y)) g(y) \,d μ(y) - \sum_{j = 1}^{ N} λ_j Φ(ρ(x, y_j)) \right| \leq C N^{- \frac{1}{2} - \frac{3}{2β}} \sqrt{\log N}, \end{align*} where the constant $C>0$ is independent of $N$ and $g$. In the case when $X$ is the unit sphere $S^d$ of $R^{d+1}$ with the ususal geodesic distance, we also prove that the constant $C$ here is independent of the dimension $d$. Our estimates are better than those obtained from the standard Monte Carlo methods, which typically yield a weaker upper bound $N^{-\frac12}\sqrt{\log N}$.