  An important question in the theory of approximate integration is to study
the conditions on the nodes $x_{k,n}$ and weights $w_{k,n}$ that allow an
estimate of the form $$ \sup_{f\in \mathcal{B}_\gamma}|\sum_k
w_{k,n}f(x_{k,n})-\int_\mathbb{X} fd\mu^*| \le cn^{-\gamma}, \qquad
n=1,2,\cdots, $$ where $\mathbb{X}$ is often a manifold with its volume measure
$\mu^*$, and $\mathcal{B}_\gamma$ is the unit ball of a suitably defined
smoothness class, parametrized by $\gamma$. In this paper, we study this
question in the context of a quasi-metric, locally compact, measure space
$\mathbb{X}$ with a probability measure $\mu^*$. We show that quadrature
formulas exact for integrating the so called diffusion polynomials of degree
$<n$ satisfy such estimates. Without requiring exactness, such formulas can be
obtained as a solutions of some kernel-based optimization problem. We discuss
the connection with the question of optimal covering radius. Our results
generalize in some sense many recent results in this direction.
