## 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

It is left as an exercise to show that the Lebesgue integral of *f(x) = 1*over the Cantor middle-third set.*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.