Folland's Advanced Calculus uses uniform convergence to justify the interchange of limits (i.e. to change order of integration and summation). But actually uniform convergence is far powerful than only justifying such an action. For example Fubini's theorem states a sufficient condition to interchange the order of limits and this condition is far weaker than uniform convergence.
Is it pedagocical thing or mentioned stronger theorems need much background than the level of Advanced Calculus? Even integrating (Riemann) the geometric series term by term $$\sum_{n=0}^{\infty}x^n$$ cannot be justified via uniform convergence since it is not uniformly convergent on $(0,1)$ since the series grows unboundedly near $1$. But Fubini's theorem states that if $f_n(x) \geq 0$ for all $x\in(0,1)$ and for all $n\in\mathbb{N}$ then the series can be integrated term by term. Why there are questions like this in Folland. I get a bit frustrated when I cannot justify the actions I conduct, properly.
Fubini's theorem is a deep result of measure theory. Folland probably uses uniform convergence in order to stay in sync with the background of the reader. The geometric series does not converge uniformly, but it converges locally uniformly (or uniformly over compact subsets, if you prefer).