Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 205 / 1 / a / ii / Solution

Every $2\times 2$ principal submatrix of a kernel matrix is positive semidefinite, soThe Cauchy-Schwarz inequality for integrals therefore givesThus every entry of $k(x,y)=\int k_{\tau }(x,y)d\tau$ is well defined. For any finite coefficients $c_r$ and points $x_r$, linearity of the integral givesbecause the integrand is nonnegative. Hence $k$ is a positive-semidefinite kernel. This proves the integral closure of positive-semidefinite kernels.