## Example 7.4.6(c): Lebesgue Integral for Bounded Functions

Is the Dirichlet function restricted to

The Dirichlet function restricted to *[0, 1]*Lebesgue integrable? If so, find the integral.*[0, 1]*is a simple function and we have already seen that its integral is zero. But we have now

*two*definitions of the integral in case of a simple function, and we need to show that both definitions agree.

If
*f(x) = c _{j} X_{Ej}*
is a simple function, then the infimum over the integrals
of all simple functions bigger than

*f*must be smaller than the integral of

*f*, which is itself a simple function. In other words, if

*f*is a simple function than we have automatically

Similarly we haveI^{*}(f)_{L}f(x) dx = c_{j}m(E_{j})

But sinceI_{*}(f)_{L}f(x) dx = c_{j}m(E_{j})

*I*we have for every simple function

_{*}(f)_{L}I^{*}(f)_{L}*f*that

so that for simple functions both definitions of the Lebesgue integral agree. In particular, the Lebesgue integral of the Dirichlet function overc_{j}m(E_{j}) I_{*}(f)_{L}I^{*}(f)_{L}c_{j}m(E_{j})

*[0, 1]*is then zero.