  In this paper, we study error bounds for {\em Bayesian quadrature} (BQ), with
an emphasis on noisy settings, randomized algorithms, and average-case
performance measures. We seek to approximate the integral of functions in a
{\em Reproducing Kernel Hilbert Space} (RKHS), particularly focusing on the
Mat\'ern-$\nu$ and squared exponential (SE) kernels, with samples from the
function potentially being corrupted by Gaussian noise. We provide a two-step
meta-algorithm that serves as a general tool for relating the average-case
quadrature error with the $L^2$-function approximation error. When specialized
to the Mat\'ern kernel, we recover an existing near-optimal error rate while
avoiding the existing method of repeatedly sampling points. When specialized to
other settings, we obtain new average-case results for settings including the
SE kernel with noise and the Mat\'ern kernel with misspecification. Finally, we
present algorithm-independent lower bounds that have greater generality and/or
give distinct proofs compared to existing ones.
