Let $E$ be a subset of the interval $[a,b]$. My question is, under what circumstances is the characteristic function $1_E$ Riemannn integrable on $[a,b]$?
Now a function is Riemann integrable if and only if its set of discontinuities is of Lebesgue measure zero. And the set of discontinuities of $1_E$ is equal to the boundary of $E$. So this is equivalent to asking, under what circumstances does the boundary of a set $E$ have measure zero? $E$ having measure zero isn't a strong enough condition, because a set of measure zero could have a boundary of positive measure. So what condition does $E$ need to satisfy?
And what is the Sigma algebra generated by sets with Riemann integrable characteristic functions?
Well lets just consider $X=\delta E$. If this contains an open set, it is obviously not integrable, since your incontinuities contain an open set, which contains an interval, and hence has positive Lebesgue measure.
In the other direction, assume the characteristic function is not Riemann integrable, i.e. your boundary does not have measure zero, and since it is closed, it is actually measurable! Hence a generating measurable set with positive measure is contained in $X$. But those generating sets are precisely the open sets!
Also, pls forgive me if I forgot some intricacies with the definition of the Lebesgue sigma algebra. been some time since I did measure theory.