Holomorphic dependence of a parameter integral

Statement

Let $F(z,x)$ be measurable in $x$ , holomorphic in $z\in\Omega$ outside a fixed null set, and suppose each compact $K\subset\Omega$ has an integrable majorant $g_K$ with $|F(z,x)|\le g_K(x)$ for all $z\in K$ . Then

H(z)=\int F(z,x)\,d\mu(x)

is holomorphic and $H^{(m)}(z)=\int\partial_z^mF(z,x)\,d\mu(x)$ .

Local justification

Use the Cauchy integral formula on a circle surrounding a smaller disc. Domination permits exchanging the contour and measure integrals. The Cauchy kernel then represents $H$ as a holomorphic function and supplies integrable bounds for every fixed derivative on the smaller disc. The original majorant must hold throughout a complex compact neighborhood; a bound only at real parameters is insufficient for this argument.