Assume you have to deal with the following derivative:
$$\frac{d}{dt}\int_{a(t)}^{b(t)}f(t,x)\mu_t(dx),$$
for some properly regular function $f,a,b$ and where $\mu_t$ is a Borel measure for each time, possibly not absolutely continuous w.r.t. Lebesgue. The question is how can one explicit this: is there any "Leibniz integral rule style" theorem from measure theory dealing with this case?