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Motivated by applications to the study of stochastic processes, we introducea new analysis of positive definite kernels $K$, their reproducing kernelHilbert spaces (RKHS), and an associated family of feature spaces that may bechosen in the form $L^{2}\left(\mu\right)$; and we study the question of whichmeasures $\mu$ are right for a particular kernel $K$. The answer to thisdepends on the particular application at hand. Such applications are the focusof the separate sections in the paper.

Reproducing kernels and choices of associated feature spaces, in the form of $L^{2}$-spaces