Certainly countability plays a role when measures are involved. Galois theory has a few such examples. For example, by a result of Jarden, for a countable field $K$, the set of $\sigma$ in the absolute Galois group $G_{K(t)}$ of the rational function field $K(t)$ for which the fixed field ${\rm Fix}(\sigma)$ is pseudo-algebraically closed (i.e. every geometrically irreducible variety over it has a dense set of rational points) has measure 1 with respect to the Haar measure on $G_{K(t)}$. Jarden-Shelah gave an example of an uncountable field $K$ where the set of $\sigma\in G_{K(t)}$ with ${\rm Fix}(\sigma)$ pseudo-algebraically closed is non-measurable.