Interchanging integrals and continuous linear forms in RKHS

Question

I am reading Reproducing kernel Hilbert spaces in probability and statistics by A Berlinet, C Thomas-Agnan.

In Chapter 5 INTEGRATION OF $\mathcal{H}$-VALUED RANDOM VARIABLES they write One of the most useful result is the possibility of interchanging integrals and continuous linear forms. however I cannot find an explicit statement on the requirements therefore.

We have the following preliminaries:

• $(E,\mathcal{T})$ a measurable space,
• $\mathcal{M}$ the space of signed measures on $(E,\mathcal{T})$,
• $\mathcal{P}$ the set of probability measures on $(E,\mathcal{T})$,
• $K$ a real reproducing kernel on $E\times E$,
• $\mathcal{H}$ the (separable) reproducing kernel Hilbert space (RKHS) with kernel $K$,
• $\mathcal{B}_\mathcal{H}$ is the Borel $\sigma$-algebra of $\mathcal{H}$

Page 210:

If $K$ is bounded and measurable then \begin{align} (E,\mathcal{T},\mu) &\to (\mathcal{H},\mathcal{B}_\mathcal{H})\\ x & \mapsto K(.,x) \end{align} is strongly (Bochner) integrable for all $\mu$ in $\mathcal{M}$ and we can define a mapping \begin{align} I\colon \mathcal{M} &\to \mathcal{H}\\ \mu &\mapsto I_\mu = \int K(.,x) d\mu(x) \end{align}

The authors write earlier on page 188 (boundness on $K$ and separability of $\mathcal{H}$ is note required):

At this stage assume that inner product and integrals can be exchanged and the validity of Fubini formula. Then for $\mu,\nu$ in $M$, \begin{align} \langle I_\mu, I_\nu\rangle = \int \left( \int K(s,t) d\nu(s)\right) d\mu(t) = \int K d(\mu \otimes \nu) \end{align}

I am interested in the conditions on $K$ such that

1. inner product and integrals can be exchanged
2. Fubini's formula is valid

Is it sufficient that $K$ is bounded and measurable? I think a key is the separability if $\mathcal{H}$.

Edit:

A Google books preview can be found here and I think I am allowed to quote some theorems if requested. Here a shorter article with similar content at page 153.