The kernel mean embedding is to map a distribution $\mathbb{P}$ to an RKHS by using the following formula $$\phi(\mathbb{P}) = \mu_{\mathbb{P}} = E_{X}(k(x,\cdot))=\int_{\mathcal{X}}k(x,\cdot)d\mathbb{P}(x). $$
In related literature, it is said that this integral should be interpreted as a Bochner integral. Can anyone elaborate this? Thanks in advance.
