Let $(X, \mathcal{B}, \mu)$ be a probability space, and let $k(\cdot, \cdot): X \times X \rightarrow \mathbb{R}$ be the Gaussian kernel. Denote $\mathcal{H}$ as the RKHS induced by $k$ on $X$. Let $f, g \in \mathcal{H}$, and I am wondering whether there is any relationship between $\int_X fg d\mu$ and $<f, g>_\mathcal{H}$.
An obvious observation is that the inner product in the RKHS is independent of the measure in the original space, which suggests that there should be no direct connection between this integral and the inner product. However, consider the following operation: let $\mu_P$ be the Kernel Mean Embedding (KME) of $P$, i.e., $\mu_P = \int_X k(x, \cdot) dP(x)$. We know that if we allow $P$ to have a square-integrable density $p$, then $\mu_P = \varepsilon_k p$, where $\varepsilon_k$ is the integral operator. This operation seems to be $\mu_P = \int_X k(x, \cdot) p(x) dx$, treating the integral of $k(x, \cdot)$ and $p(x)$ as an operation similar to the inner product. I am wondering if all functions $f$ satisfy $\int_X k(x, \cdot) f dx = f(x)$?