Let μ be a Borel probability measure on a compact path-connected metric space $(X,\rho )$ for which there exist constants $c,\beta \ge 1$ such that $\mu (B)\ge c{r}^{\beta }$ for every open ball $B\subset X$ of radius $r>0$. For a class of Lipschitz functions $\mathrm{\Phi }:[0,\infty )\to \mathbb{R}$ that are piecewise within 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 $N\ge 2$, and each $g\in L^{\infty }(X,d\mu )$ with ${\Vert g\Vert }_{\infty }=1$, there exist points $y_1,\dots ,y_N\in X$ and real numbers ${\lambda }_1,\dots ,{\lambda }_N$ such that for any $x\in X$,$|\underset{X}{\int }\mathrm{\Phi }(\rho (x,y))g(y)\mathrm{d}\mu (y)-\sum _{j=1}^N{\lambda }_j\mathrm{\Phi }(\rho (x,y_j))|⩽C{N}^{-\frac{1}{2}-\frac{3}{2\beta }}\sqrt{\log N},$ where the constant $C>0$ is independent of N and g. In the case when X is the unit sphere ${\mathbb{S}}^d$ of ${\mathbb{R}}^{d+1}$ with the usual 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 $N^{-\frac{1}{2}}\sqrt{\log N}$.

令μ为紧路径连接度量空间上的Borel概率度量$（X，\rho ）$ 对于存在常数 $C，\beta \ge \mathrm{1个}$ 这样 $\mu （乙）\ge C{\mathrm{[R}}^{\beta }$ 每个开球 $乙\subset X$ 半径 $\mathrm{[R}>0$。对于一类Lipschitz函数$\mathrm{\Phi }：[0，\infty ）\to \mathbb{[R}$在连续函数的有限维子空间中是分段的，我们证明在某些温和条件下，度量ρ和度量μ对于每个正整数$ñ\ge 2$，以及每个 $G\in {\mathrm{大号}}^{\infty }（X，d\mu ）$ 与 ${\Vert G\Vert }_{\infty }=\mathrm{1个}$，存在点 ${ÿ}_{\mathrm{1个}}，\dots ，{ÿ}_{ñ}\in X$ 和实数 ${\lambda }_{\mathrm{1个}}，\dots ，{\lambda }_{ñ}$ 这样对于任何 $X\in X$，$|\underset{X}{\int }\mathrm{\Phi }（\rho （X，ÿ））G（ÿ）\mathrm{d}\mu （ÿ）-\sum _{Ĵ=\mathrm{1个}}^{ñ}{\lambda }_{Ĵ}\mathrm{\Phi }（\rho （X，{ÿ}_{Ĵ}））|⩽C{ñ}^{-\frac{\mathrm{1个}}{2}-\frac{3}{2\beta }}\sqrt{\mathrm{日志}ñ}，$ 常数 $C>0$独立于N和g。在X是单位球体的情况下${\mathbb{小号}}^d$ 的 ${\mathbb{[R}}^{d+\mathrm{1个}}$在通常的测地距离下，我们还证明了此处的常数C与维数d无关。我们的估算结果要好于从标准蒙特卡洛方法获得的估算结果，后者通常会产生较弱的上限${ñ}^{-\frac{\mathrm{1个}}{2}}\sqrt{\mathrm{日志}ñ}$。