Let consider $F:U \to \mathbb{C}, z \mapsto \int \limits_I f(z,t)\mathrm{d}t$ where $U$ is an open set of $\mathbb{C}$ and $I$ an interval of $\mathbb{R}$.
Here is the statement :
If for all $z \in U$, the function $x\mapsto f(t,z)$ is measurable on $I$.
If for all $t \in I$, the function $z\mapsto f(t,z)$ is holomorphic on $U$.
If for all $z \in U$ and $t\in I$, we have $\mid f(t,z) \mid \le \phi(t)$ where $\int \limits_I\mid \phi(t)\mid\mathrm{d}t <+\infty.$
Then $F$ is holomorphic on $U$.
Note that instead of $\forall t \in I$ it holds for the condition "a.e".
But I was wondering if it was not necessary to have the domination for $K \subset U$ where $K$ is compact ?
Thanks in advance !