## Proposition 7.4.14: Bounded Measurable Functions are Integrable

If

*f*is a bounded function defined on a measurable set*with finite measure. Then***E***f*is measurable if and only if*f*is Lebesgue integrable.
If *f* is a bounded function defined on a measurable set **E**
with finite measure. Then *f* is measurable if and only if
*f* is Lebesgue integrable.