I have three queries:
(1) Do term by term integration always imply the order of integral-summation, integral-limit are interchangable and vice-versa?
(2) Suppose $\{f_n\}$ be a boundedly convergent sequence of Riemann integrable functions on a set $[a, b]$. Assume that the limiting function is also Riemann integrable on $[a, b]$. If $\{f_n\}$ is not uniformly convergent on any subset of $[a, b]$, can we perform term by term integration in this case?
The second one I think have a positive response due to Arzela Theorem that states that:
Assume that $\{f_n\}$ is boundedly convergent on $[a, b]$ and suppose each $f_n$ is Riemann-integrable on $[a, b]$. Assume also that the limit function $f$ is Riemann-integrable on $[a, b]$. Then $$\lim_{n \to \infty} \int_{a}^{b} f_n (t)~dt = \int_{a}^{b} \lim_{n \to \infty} f_n (t)~dt = \int_{a}^{b} f(t)~dt.$$
(3) Can we use the Arzela's theorem for series of functions as well? In other words, can we perform term by term integration for non-uniformly convergent but boundedly convergent series of functions?