## 7.4. Lebesgue Integral

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

Is the function

Let *f(x) = x*Lebesgue integrable over the rational numbers inside^{2}*[0, 2]*? If so, find the integral.*be the rational numbers inside*

**Q***[0, 2]*and define the functions

s(x) = 0

S(x) = 4 X_{Q}(x)

Then
*s(x) f(x) S(x)*
over * Q*, and both

*s*and

*S*are simple functions. Therefore

andI^{*}(f)_{L}S(x) dx = 4 m(Q) = 0

Since alsoI_{*}(f)_{L}s(x) dx = 0

*I*we have that

_{*}(f)_{L}I^{*}(f)_{L}*I*= 0. Therefore the function is integrable and the value of the integral is zero.

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