In Folland's book Real Analysis, he proves the following theorem (Theorem 2.27) concerning differentiation under integral sign.
Suppose $f:X \times [a,b] \to C$ is a complex valued function such that $f(\cdot,t)$ is integrable for each $t \in [a,b]$. Let $F(t)=\int_X f(x,t) d\mu(x)$. Then, if $\partial t f$ is dominated by an integrable function, we can differentiate $F(t)$ under the integral sign.
He then goes on to saying that if the parameter $t$ varies over a non-compact interval, we can still apply this theorem to any compact subinterval. But, my question is why is the theorem stated for compact interval $[a,b]$ in the first place? I do not see where/how the compactness of the interval is used in the proof