On reproducing kernels and equivalent measures

Question

Let $\mathcal{H}$ be a reproducing kernel Hilbert space with kernel $k$ on some Hilbert space $L^2(\mathbb{R}, \mu)$. Suppose there exists another measure $\nu$ that is equivalent to $\mu$, i.e. $\mu \ll \nu$ and $\nu \ll \mu$. Let's say I view $\mathcal{H}$ as a subspace of $L^2(\mathbb{R}, \nu)$ using the Hilbert space isomorphism induced by the Radon-Nikodym theorem. Does the above Hilbert space isomorphism preserve the structure of $\mathcal{H}$ as a reproducing kernel Hilbert space, and if so, what is its kernel?