When is a Lebesgue integrable function a Riemann integrable function?

Question

When is a Lebesgue integrable function a Riemann integrable function ?

And if we have $f\in \mathcal{L}^1([0,1],\lambda)$, does it implies that $f$ is Riemann integrable, and why ?

Answer 1

No, it doesn't.

It was proven by Lebesgue that a bounded function $f$ is Riemann integrable precisely when the set of points where $f$ is discontinuous has measure zero.

Answer 2

This is the Riemann-Lebesgue Theorem, which says that $f$ is Riemann integrable if and only if the set of discontinuity points is of measure zero.

Note that $1_{\mathbb{Q}}$ (the indicator of the rationals) is Lebesgue-integrable but not Riemann integrable. On $[0,1]$ it is almost everywhere bounded but not continuous. Here, the set of discontinuity points is of measure $1$. So we cannot say anything about continuity and Lebesgue integrability (except that every continuous function on a set of finite measure is Lebesgue-integrable).