Integral closure of positive-semidefinite kernels
= 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 $2\times2$ <kernel matrix> gives
$$
|k_\tau(x,y)|\leq\sqrt{k_\tau(x,x)k_\tau(y,y)}.
$$
The <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.