Integral closure of positive-semidefinite kernels

Suppose $k_{\tau }$ is a positive-semidefinite kernel for every $\tau$ and its diagonal entries are integrable. Positivity of each $2\times 2$ kernel matrix givesThe Cauchy-Schwarz inequality makes every entry integrable, and integrating each nonnegative finite quadratic form shows that $\int k_{\tau }d\tau$ is again a positive-semidefinite kernel.