An extension of Mercer theorem to matrix-valued measurable kernels

We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space X into C-n. Given a finite measure mu on X, we represent the reproducing kernel K as a convergent series in terms of the eigenfunctions of a suitable compact operator depending on K and mu. Our result holds under the mild assumption that K is measurable and the associated Hilbert space is separable. Furthermore, we show that X has a natural second countable topology with respect to which the eigenfunctions are continuous and such that the series representing K uniformly converges to K on compact subsets of X x X, provided that the support of mu is X. (C) 2012 Elsevier Inc. All rights reserved.