## 7.4. Lebesgue Integral

### Definition 7.4.5: Lebesgue Integral for Bounded Functions

Suppose

*f*is a bounded function defined on a measurable set*with finite measure. Define the***E***upper*and*lower*Lebesgue integrals, respectively, asIfIis simple and^{*}(f)_{L}= inf{ s(x) dx: ss f }

Iis simple and_{*}(f)_{L}= sup{ s(x) dx: ss f }

*I*the function^{*}(f)_{L}= I_{*}(f)_{L}*f*is called Lebesgue integrable over*and the Lebesgue integral of***E***f*over*is denoted by***E**_{E}f(x) dx