  In this paper, we prove convergence and fluctuation results for
measure-valued P\'olya processes (MVPPs, also known as P\'olya urns with
infinitely-many colours). Our convergence results hold almost surely and in
$L^2$, under assumptions that are different from that of other convergence
results in the literature. Our fluctuation results are the first second-order
results in the literature on MVPPs; they generalise classical fluctuation
results from the literature on finitely-many-colour P\'olya urns. As in the
finitely-many-colour case, the order and shape of the fluctuations depend on
whether the "spectral gap is small or large".
  To prove these results, we show that MVPPs are stochastic approximations
taking values in the set of measures on a measurable space $E$ (the colour
space). We then use martingale methods and standard operator theory to prove
convergence and fluctuation results for these stochastic approximations.
