Suppose that $f_n(t;z)\to f(t;z)$ with $z$ a complex parameter, such that $$g(z) = \int_Af(t;z)dt,$$ and $$g(z)=\lim_{n\to\infty}\int_Af_n(t;z)dt,$$ hold, where $A$ is some real interval, e.g. $(1,3)$, $(1,\infty)$, etc.
Do these conditions imply $\int_A f_n\to g$ uniformly ? Or maybe this is not true.