I am looking for a citable reference for the result on differentiation under the integral sign for integration against a measure.
The result states that if $R \subset \mathbb R$, $(X,\mathcal F, \mu)$ is a measure space, and $f: R \times X \rightarrow \mathbb{R}$ satisfies:
1) $x \mapsto f(t,x)$ is integrable for all (fixed) $t \in R$.
2) $t \mapsto f(t,x)$ is almost everywhere differentiable for (fixed) $x \in X$.
3) There exists integrable $g: X \rightarrow \mathbb R$ such that
$$ | \partial_t f(t,x) | \leq g(x) $$
Then the function
$$F(t) = \int_X f(t,x) \mu(dx) $$
is differentiable with derivative
$$F'(t) = \int_X \partial_t f(t,x) \mu(dx) $$
Best Answer
I found the answer to your question on p.142 of "Probability Theory: A Comprehensive Course, Ed. 2" by Klenke. Its called the Differentiation Lemma in that text. You've stated, almost word for word, the pre-conditions of that lemma. Here is a link that describes the conditions on exchanging integration and diffeentiation.