Interactive Real Analysis
- part of Example 7.4.6(d): Lebesgue Integral for Bounded Functions
Is every bounded function Lebesgue integrable?
Context
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).
ContextWe 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 A such that g -1 (A) is not measurable. That function turns out to be a bounded function which is not Lebesgue integrable.