## 7.1. Riemann Integral

### Theorem 7.1.16: Lebesgue's Theorem

f

*f*is a bounded function defined on a closed, bounded interval*[a, b]*then*f*is Riemann integrable if and only if the set of points where*f*is discontinuous has measure zero.As we will see later, any set of finitely or countably many points is a set of measure zero.

To prove this theorem, we would need to know more about measure theory, which at this point we do not. So, we will postpone this proof.