  Let $\mu$ be a Borel probability measure on a compact path-connected metric
space $(X, \rho)$ for which there exist constants $c,\beta>1$ such that $\mu(B)
\geq c r^{\beta}$ for every open ball $B\subset X$ of radius $r>0$. For a class
of Lipschitz functions $\Phi:[0,\infty)\to R$ that piecewisely lie in a
finite-dimensional subspace of continuous functions, we prove under certain
mild conditions on the metric $\rho$ and the measure $\mu$ that for each
positive integer $N\geq 2$, and each $g\in L^\infty(X, d\mu)$ with
$\|g\|_\infty=1$, there exist points $y_1, \ldots, y_{ N}\in X$ and real
numbers $\lambda_1, \ldots, \lambda_{ N}$ such that for any $x\in X$,
  \begin{align*}
  & \left| \int_X \Phi (\rho (x, y)) g(y) \,d \mu (y) - \sum_{j =
  1}^{ N} \lambda_j \Phi (\rho (x, y_j)) \right| \leq C N^{- \frac{1}{2} -
\frac{3}{2\beta}} \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 $N^{-\frac12}\sqrt{\log N}$.
