7.4. Lebesgue Integral
Example 7.4.9(c): Riemann implies Lebesgue Integrable
It does not make sense to ask for the Riemann integral of a function defined over a set that is not an interval, so there's nothing to do for the second part.
But we could rephrase the question: take the function f that is equal to 1 over the Cantor set and zero everywhere else and find the Riemann integral of that function over some interval. Now it is a well-defined question, but the Riemann integral does not exist, which is - again (!) - left as an exercise.