Is it true that a bounded function $ f:[a,b]\to \mathbb{R} $ is Lebesgue integrable if and only if there exists a function $f_1:[a,b]\to \mathbb{R}$ that is Riemann integrable and $$ m^*\lbrace x: (f-f_1)(x)\ne 0\rbrace =0 $$
One side of the above assumption is true and obvious.
No. Take $f=1_A$, where $A$ is a measurable set with the property that both $A$ and $A^c$ have positive measure intersection with any interval (See for example, Construction of a Borel set with positive but not full measure in each interval). It is not too hard to see that modification on a null set $N$ cannot make such an $1_A$ continuous a.e. (because $A\setminus N$ and $A^c\setminus N$ must accumulate at every point).
