TITLE: When can we move a Fréchet derivative under a Lebesgue integral?
QUESTION [5 upvotes]: Under what conditions can we move a Fréchet derivative under a Lebesgue integral?  Specifically, when does
$$
G'(x) = h\in X\mapsto \int_{\Omega} \left(F_x^\prime(x,t)h\right) \mu(dt)
$$
where
$$
G(x)=\int_{\Omega} F(x,t) \mu(dt)?
$$
I suspect this has something to do with the Dominated Convergence Theorem.  What's causing me issues, though, is the definition of a Fréchet derivative.  If we were to use a Gatteaux derivative, I can see how this works because we start with the expression for $G$, write out the definition of a directional derivative, and then move the limit to the inside of the integral.  With a Fréchet derivative, the limit sits outside of a norm expression, so it's not clear to me how to use the Dominated Convergence Theorem.

REPLY [4 votes]: I hope it's not too late to answer your question (and that it's not a duplicate). 
You can find the full proof (fool's proof? ;) ) here.
The trick has three parts:

Show that the required expression is the Frechet derivative by calculating the limit.
Because the numerator of the limit we're trying to calculate is in $\mathbb{R}$, the norm is in fact an absolute value and we can move the denominator inside.
We need some kind of mean value theorem for Fréchet. I used one from 


Ambrosetti, A. and Prodi, G. (1995). A Primer of Nonlinear Analysis. Cambridge Studies in Advanced Mathematics. Cambridge University Press. Theorem 1.8, p. 13.

I'm definitely not an expert on Fréchet derivatives (only learned about them over the weekend), so there is probably a stronger one than what I used. 
I think the integrability conditions I stated are the weakest possible. I think one has to assume the integrability of the derivatives, but I didn't construct a counter-example.