Give an example of a non-Riemann integrable function $g$ which is equal to a Riemann integrable function $f$ almost everywhere in $[a, b]$

Question

So I am having problems facing this exercise to find an example of such function.

Let $f\in R([a, b])$. Can we find a function $g\colon [a,b]\to \mathbb R$ such that $f=g$ almost everywhere and $g$ not Riemann integrable?

Answer 1

Define, $f:[a,b]\to\mathbb{R}$ by, $f(x)=0$ for all $x\in [a,b]$. Then $f$ is clearly R-integrable on $[a,b]$. Now define, $g:[a,b]\to\mathbb{R}$ by, $$ g(x)= \begin{cases} 0\quad\text{if $x$ is irrational}\\ 1\quad\text{if $x$ is rational} \end{cases} $$ Then, $g=f$ almost everywhere but note that $g$ is not R-integrable on $[a,b]$.