7.4. Lebesgue Integral
Example 7.4.2(d): Simple Functions
Is the function that is equal to 1 if x is part of the Cantor middle-third set and 0 otherwise a simple function?If C is the Cantor set, then the question is whether the characteristic function XC(x) of C is simple.
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 XC(x) is a simple function.