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.