## 7.4. Lebesgue Integral

### Theorem 7.4.7: Lebesgue Integral of Bounded Function

If

*f*is a bounded function defined on a measurable set*E*with finite measure. Then*f*is integrable if the setsforE_{k}= { x E: (k-1) M / n ≤ f(x) < k M/n }

*k = -n, -(n-1), ... 0, 1, 2, ..., (n-1), n*are measurable, then*f*is Lebesgue integrable.
**Proof:**
TBD shortly