As the title says, for every Lebesgue integrable function $f:\mathbb{R}\to\mathbb{R}$ is there a Riemann integrable function $g:\mathbb{R}\to\mathbb{R}$ such that $f=g$ almost everywhere?


For example, $\chi\_{\mathbb{Q}\cap [0, 1]}$ is a well known example of lebesgue integrable function that is not riemann integrable, but $\chi\_{\mathbb{Q}\cap [0, 1]}=0$ almost everywhere and 0 is a riemman integrable function.



Equivalence between Lebesgue integrable and Riemann integrable functions
Let $A$ be a measurable subset of $[0,1]$ such that both it and its complement have positive measure in every open interval in $[0,1]$ (see [here](https://math.stackexchange.com/questions/57317/construction-of-a-borel-set-with-positive-but-not-full-measure-in-each-interval) for example). Its characteristic function is dominated by $1\_{[0,1]}$, so it is Lebesgue integrable, but it is discontinuous everywhere on $[0,1]$ even after being modified on a null set, so it is not a.e. equal to a Riemann integrable function.


