According to Lebesgue’s characterization of Riemann integrable functions (see: http://www.math.ru.nl/~mueger/Lebesgue.pdf) cantor funciton on $[0,1]$ (https://en.wikipedia.org/wiki/Cantor_function) is riemann integrable, and since it's bounded. They should be equal. However, since $f$ (the cantor function) is singular on $[0,1]$, $f'=0$ almost everywhere. Thus remmann and lebesgue both equal to $0$.
So both lebesgue and riemann integralbe can not deal with cantor's function or Weierstrass function on $[0,1]$(https://en.wikipedia.org/wiki/Weierstrass_function).
My question was what kind of analysis techniques can be used to deal with cantor funciton, or even Weierstrass function?