## 7.4. Lebesgue Integral

### Example 7.4.2(d): Simple Functions

Is the function that is equal to 1 if

If *x*is part of the Cantor middle-third set and 0 otherwise a simple function?*is the Cantor set, then the question is whether the characteristic function*

**C***X*of

_{C}(x)*is simple.*

**C**
By definition, a characteristic function is simple if the underlying set
is measurable. Since we have shown before that * C* is measurable
(with measure zero), the function

*X*is a simple function.

_{C}(x)