7.4. Lebesgue Integral

Example 7.4.7(c): Riemann implies Lebesgue Integrable

If possible, find the Riemann and Lebesgue integrals of the constant function f(x) = 1 over the Cantor middle-third set.
It is left as an exercise to show that the Lebesgue integral of f over the Cantor set is zero.

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.

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