Stochastic Fubini

Question

Provided a stochastic basis $(\Omega,\mathfrak A,\mathfrak F,P)$ supporting a continuous local martingale $M$ with time horizon $T>0$, and a measure space $(E,\mathfrak E,\mu)$, let $F: \mathbb [0,T] \times E \times \Omega \rightarrow \mathbb R, (t,e,\omega) \mapsto F_t(e,\omega)$ be a map such that:

1. $F(.,.,e)$ is predictable and $P(\int_0^\infty F_t(e)^2~d\langle M \rangle_t <\infty) =1$, for any $e\in E$,
2. $e\mapsto \int_0^T F_t(e)~dM_t \in L^1(\mu)$,
3. $e\mapsto F_t(e,\omega) \in L^1(\mu)$ for any $(t,\omega) \in [0,T] \times \Omega$,
4. $(t,\omega)\mapsto \int_E F_t(e,\omega) ~d\mu(e)$ is predictable and $P\Big(\int_0^T \Big(\int_E F_t(e) ~d\mu(e)\Big)^2~d\langle M\rangle_t<\infty\Big) = 1$.

Under which conditions does the following statement hold: \begin{equation*} \int_0^T \Big(\int_E F_t(e) ~d\mu(e)\Big)~dM_t = \int_E \Big(\int_0^T F_t(e) ~dM_t\Big)~d\mu(e)\quad ? \end{equation*}

I am primarily concerned with the case of $M$ being a Wiener process and $E$ being a compact interval in $\mathbb R$ with Lebesgue measure.

Idea: For simple functions, this seems to hold. So by a monotone class argument and localisation, one may arrive at the conclusion.

Question: I did not find information about this in some of the standard references on stochastic calculus I use. Could you recommend me some where I can find information about this issue?