## Example 7.4.6(d): Lebesgue Integral for Bounded Functions

That would be too nice to be true - therefore it is not true! But it is difficult to find a bounded function that is not Lebesgue integrable (whereas it is easy to find a bounded function that is not*Riemann*integrable).

We have said before that we can prove that a bounded function with the property that the inverse image of a measurable set is measurable would be Lebesgue integrable. To find a bounded function that is not integrable we therefore need to find a function for which that property is not true.

If *C(x)* is the
Cantor function defined in chapter 6,
then let *f(x) = C(x) + x*. It can be shown that
*f* has bounded inverse function *g = f ^{ -1 }*
and that there exists a measurable set

*such that*

**A***g*is not measurable. That function turns out to be a bounded function which is not Lebesgue integrable.

^{ -1 }(*)***A**