Is there always a Riemann integrable $g$ for a Lebesgue integrable $f$ so that $f = g$ a.e.

Question

The following is an exercise from Bruckner's Real Analysis:

Prove or disprove that if $f$ is a bounded function and Lebesgue integrable on an interval $[a, b]$, then there exists a Riemann integrable function $g$ so that $f = g$ a.e. and $\int_{[a,b]} f dλ= \int_a^b g(x) dx$.

I checked through the theorems and examples of the book and also MSE, neither could I find a counter-example nor a proof if it's a theorem. Detailed explanation as well as hints would be much appreciated.

Answer 1

Hint:

The indicator function of a Fat Cantor set $F$ may be a candidate. It is discontinuous at every point in the Cantor set (by virtue of Cantor sets not containing any open interval).