## 7.4. Lebesgue Integral

### Example 7.4.10(a): Properties of the Lebesgue Integral

Is the function

Define the function *f(x) = x e*Lebesgue integrable over the Cantor middle-third set? If so, find the integral.^{x}*g(x) = 0*on

*[0, 1]*. Then

*f = g*except on the Cantor middle third set

*. Since*

**C***m(*the functions

*) = 0***C***f*and

*g*agree except on a set with measure zero.

Therefore the integrals of *f* and *g* also agree,
but clearly

so that_{[0, 1]}g(x) dx = 0

as well_{C}f(x) dx = 0

We may have skipped a step to go from the integral over *[0, 1]*
to the integral over * C*, but the details are easily filled in.