Karhunen-Lo\`eve expansion of Random Measures

We present an orthogonal expansion for real, function-regulated, second-order random measures over $\mathbb{R}^{d}$ with measure covariance. Such a expansion, which can be seen as a Karhunen-Lo\`eve decomposition, consists in a series of deterministic real measures weighted by uncorrelated real random variables with the variances forming a convergent series. The convergence of the series is in a mean-square sense stochastically and against measurable bounded test functions (with compact support if the random measure is not finite) in the measure sense, which implies set-wise convergence. This is proven taking advantage of the extra requirement of having a covariance measure over $\mathbb{R}^{d}\times\mathbb{R}^{d}$ describing the covariance structure of the random measure, for which we also provide a series expansion. These results cover for instance the cases of Gaussian White Noise, Poisson and Cox point processes, and can be used to obtain expansions for trawl processes.