Differentiation under the integral

Let be open and suppose that satisfies:

1. is integrable for all
2. is differentiable for all
3. for some integrable and for all ,

\left\lvert \frac{\partial f}{\partial t}(t,x) \right\rvert \leq g(x)

F(t)=\int_{E} f(t,x) \mu(dx)

\frac{d}{dt} F(t)=\int_{E} \frac{\partial f}{\partial t}(t,x)\mu(dx)