  Let ${\mathbb X}$ be a compact, connected, Riemannian manifold (without
boundary), $\rho$ be the geodesic distance on ${\mathbb X}$, $\mu$ be a
probability measure on ${\mathbb X}$, and $\{\phi_k\}$ be an orthonormal system
of continuous functions, $\phi_0(x)=1$ for all $x\in{\mathbb X}$,
$\{\ell_k\}_{k=0}^\infty$ be an nondecreasing sequence of real numbers with
$\ell_0=1$, $\ell_k\uparrow\infty$ as $k\to\infty$, $\Pi_L:={\mathsf
{span}}\{\phi_j : \ell_j\le L\}$, $L\ge 0$. We describe conditions to ensure an
equivalence between the $L^p$ norms of elements of $\Pi_L$ with their suitably
discretized versions. We also give intrinsic criteria to determine if any
system of weights and nodes allows such inequalities. The results are stated in
a very general form, applicable for example, when the discretization of the
integrals is based on weighted averages of the elements of $\Pi_L$ on geodesic
balls rather than point evaluations.
