= Dominated differentiation under the integral sign
{title2=$F'(t)=\int_X\partial_t h(t,x)\,d\mu(x)$}
Let $t_0$ be interior to a parameter interval. Suppose $h(t,\cdot)$ is a <measurable function> for each $t$, $h(t_0,\cdot)$ is a <Lebesgue integrable function>, and $h(\cdot,x)$ is differentiable near $t_0$ for $x$ <almost everywhere>. If one integrable nonnegative function $D$ satisfies $|\partial_t h(t,x)|\leq D(x)$ throughout that neighbourhood off one fixed null set, then $F(t)=\int_Xh(t,x)\,d\mu(x)$ is defined there and satisfies the displayed derivative formula at $t_0$.
The <mean value theorem> gives $|h(t,x)-h(t_0,x)|\leq|t-t_0|D(x)$, proving local integrability. It also bounds the difference quotient by $D$. That quotient converges <almost everywhere> to $\partial_t h(t_0,x)$, so the <dominated convergence theorem> permits passing the limit through the <Lebesgue integral>.
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