Let $E$ be a closed nowhere dense set on $[0,1]$.
a) Is it true that $\chi_E$ is Lebesgue integrable?
b) is it true that $\chi_E$ is Riemann integrable if and only if $m(E)=0$?
I know that $f$ is Riemann integrable if and only if $f$ is continuous except on a set of measure 0.
By definition of nowhere dense – the interior of the closure is empty – I think that b) is true. But not sure about a).
Also, please let me know if you think my reasoning for b) is incorrect.
Thanks.
Best Answer
You did not give details for b). Here is a proof: If $m(E)=0$ then $\chi_E$ is continuous a..e because it is continuous on $E^{c}$ (It is the constant function $1$ on this open set). Hence, it is RI. Conversely, if it is RI then it is continuous a.e.. If $x \in E$ then no interval around $x$ can be conatined in $E$ (becasue $E$ has no interior) and hence there is a sequence $x_n$ in $E^{c}$ converging to $x$. But then $\chi_E (x_n) =0$ for all $n$ and $\chi_E(x)=1$. So $\chi_E$ is not continuous at any point $x$ of $E$. Since it is continuous a.e we must have $m(E)=0$.