I'm currently working on a project in which we define a new type of integral. And I'm trying to intechange the integral with expectation, something like $\mathbb{E} \left[ (\mathcal{N})\int f dW \right]=(\mathcal{N})\int \left[ \mathbb{E} f dW \right]$, where $\mathcal{N}$ denoted the defined integral, $f$ is an operator-valued stochastic process and $W$ a Hilbert-space valued $Q$-Weiner Process. Under which conditions can I do such thing?
What I give might be vague. But, could this involve something like Fubini-Tonelli Theorem? I haven't also read a version of this theorem in for function in infinite dimensions. If there is, please cite. Thank you in advance! :)
It seems that the Theorem you need is Theorem 2 in Chapter X in the book The Bochner Integral isbn 9780124958500