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?