While reading Real Analysis- Stein, Shakarchi, I came up with following questions, some of which are in the "Introduction" of the book. The purpose of a series of questions is "to get actual introduction" to further study the subject in a proper direction. Since I was working in Algebra, and switching to Analysis now, I am not too familiar with Fourier series, Lebesgue Measure, Integration etc.
1) If $f$ is Riemann integrable on $[-\pi,\pi]$, and is periodic, the there is a Fourier series associated to it - $\sum_{n=-\infty}^\infty a_ne^{inx}$, and the Fourier coefficients $a_i$ satisfy that $\sum_{i}|a_i|^2=\frac{1}{2\pi} \int_{-\pi}^\pi |f(x)|^2dx$. In particular, the series $\sum |a_i|^2$ converges. Conversely, given a sequence $\{a_i\}$ such that $\sum |a_i|^2$ is convergent, what is the nature of the function defined by Fourier series $\int a_ne^{inx}$? It may not be Riemann integrable (I think, and I don't know details), but is it Lebesgue Integrable? Does the Fourier series converges almost everywhere?
2) There is a sequence of Riemann integrable functions on $[0,1]$ such that the equality $\lim \int f_n= \int \lim f_n$ doesn't hold. So, what are the functions $f_n,f$ on $[0,1]$ such that the above equality holds? How Lebesgue theory answers this question?
3) If $\Gamma=(x(t),y(t))$, $a\leq t\leq b$, is a rectifiable curve then the length of the curve is expected to be $\int_{a}^b (x'(t)^2+y'(t)^2)^{1/2}dt$. But this equality does not hold in general (as Stein-Shakarchi says). How Lebesgue theory gives a good definition of length of the curve?