Author-hosted lecture notes: www.gatsby.ucl.ac.uk/~gretton/coursefiles/RKHS_Notes1.pdf
The title page gives both authors and 11 March2012. Consulted passage: section6.2, Theorem48, printed p21: continuous kernel on a compact metric space, finite Borel measure of full support, positive-eigenvalue expansion, uniform convergence and pointwise absolute convergence. The surrounding text fixes a positive-definite kernel and chooses continuous representatives of eigenfunctions.