Let $\mu$ be a finite Borel measure on a non-empty Borel subset $X\subseteq (0,\infty)$. The space $H\subseteq C(X;\mathbb{R})$ such that every element of $H$ can be identified with an element of $ L_{\mu}^2(X)$; where $X$ is equipped with the relative topology on $X$ induced by restriction from $(0,\infty)$.
Fix a distinguised point $x^{\star}\in X$. If we define the inner-product on $H$ to be $$ \left\langle f,g \right\rangle\triangleq f(x^{\star})g(x^{\star}) + \int_{x \in X} f(x)g(x) \, d\mu(x), $$ then does $H$ admit an (infinite-dimenional) RKHS subspace?