Differentiation under an integral sign

Statement

Let $I$ be an open real interval and $F(t,x)$ measurable in $x$ . Suppose outside a fixed null set, $F(\cdot,x)$ is $C^1$ on $I$ , and $F(t_0,\cdot)\in L^1(X,\mu)$ for some $t_0\in I$ . Suppose each compact subinterval $J\subset I$ has an integrable majorant $g_J$ with $|\partial_tF(t,x)|\le g_J(x)$ for all $t\in J$ . Then

\frac{d}{dt}\int_X F(t,x)\,d\mu(x)=\int_X\partial_tF(t,x)\,d\mu(x).

Justification and higher orders

The mean value theorem bounds a difference quotient by a majorant on a slightly larger parameter interval. Dominated convergence then passes its limit through the integral. Applying the same argument to derivatives proves the $C^k$ version when derivatives through order $k$ have local integrable majorants. A moving integration domain contributes boundary terms and is not covered by this fixed-domain statement.