I have read the following theorem from Apostol's analysis.
Let $\alpha$ be of bounded variation on $[a, b]$. Assume that each term of the sequence $\{f_n\}$ is real valued function such that $f_n \in R(\alpha)$ on $[a, b]$ for each $n = 1, 2, \ldots$ Assume that $f_n \to f$ uniformly on $[a, b]$ and define $g_n (x) = \int_{a}^{x} f_n (t)~ d\alpha(t)$ if $x \in [a, b], ~~n=1, 2, \ldots$ Then we have:
(a) $f \in R(\alpha)$ on $[a, b]$.
(b)$g_n \to g$ uniformly on $[a, b]$, where $g(x)= \int_{a}^{x} f(t) ~d \alpha(t).$
My first curiosity is: whether any integrable function is again integrable for finitely many times (or infinitely many times). If so, then we can use the above fact to construct new sequence of functions again and again. The uniform convergence over some set is confirmed by the $(a)$ part of the above theorem. Hence, we are able to conclude $\int_{a}^{x} \int_{a}^{x}\ldots\int_{a}^{x} f_n \to \int_{a}^{x} \int_{a}^{x}\ldots\int_{a}^{x} f$ on some set. Here, integration is taken finitely many times. Is the observation is correct?