When are signed measures isomorphic to a RKHS?

Question

Let $\mathcal{H}$ be a reproducing kernel Hilbert space of functions over $X$, with a bounded kernel $\mathcal{K}: X\times X\to \mathbb{R}$. Let us assume there is a sigma-algebra over $X$ such that $\mathcal{K}$ is measurable in its first argument. It can be shown that for any signed measure $\mu$, $\mathcal{K}_{\mu} = \mathbb{E}_{x\sim\mu}\mathcal{K}(x, \cdot)\in\mathcal{H}$ (https://arxiv.org/pdf/1605.09522.pdf, Lemma 3.1). My question is, is $\mathcal{H} = \left\{\mathcal{K}_{\mu}: \textrm{$\mu$ is a signed measure}\right\}$? If not, what is a counterexample? Thanks!