## Example 7.4.4(c): Lebesgue Integral for Simple Functions

Find the Lebesgue integral of the Dirichlet function restricted to

If *[0, 1]*and of the characteristic function of the Cantor middle-third set.*is the set of all rational numbers, then the Dirichlet function restricted to*

**Q***[0, 1]*is the characteristic function of

*. The set*

*=***A***[0, 1]***Q***is a subset of*

**A***, hence*

**Q***is measurable and*

**A***m(*. Therefore:

*) = 0***A**The Cantor middle-third setX_{A}(x) dx = m(A) = 0

*is also measurable with measure 0. Therefore, by definition*

**C**Note that the Riemann integral of either function does not exist (which we have shown for the Dirichlet function, and leave as an exercise for the characteristic function of the Cantor set).X_{C}(x) dx = m(C) = 0